The climate system and the second law of thermodynamics
TThe climate system and the second law of thermodynamics
Martin S. Singh ∗ School of Earth, Atmosphere,and Environment,Monash University, Victoria,Australia
Morgan E O’Neill
School of Earth, Energy,and Environmental Sciences,Stanford University, California,USA (Dated: February 4, 2021)
The second law of thermodynamics implies a relationship between the net entropy exportby the Earth and its internal irreversible entropy production. The application of thisconstraint for the purpose of understanding Earth’s climate is reviewed. Both radiativeprocesses and material processes are responsible for irreversible entropy production inthe climate system. Focusing on material processes, an entropy budget for the climatesystem is derived which accounts for the multi-phase nature of the hydrological cycle.The entropy budget facilitates a heat-engine perspective of atmospheric circulationsthat has been used to propose theories for convective updraft velocities, tropical cycloneintensity, and the atmospheric meridional heat transport. Such theories can only besuccessful, however, if they properly account for the irreversible entropy productionassociated with water in all its phases in the atmosphere. Irreversibility associatedwith such moist processes is particularly important in the context of global climatechange, for which the concentration of water vapor in the atmosphere is expected toincrease, and recent developments toward understanding the response of the atmosphericheat engine to climate change are discussed. Finally, the application of variationalapproaches to the climate and geophysical flows is briefly reviewed, including the use ofequilibrium statistical mechanics to predict behavior of long-lived coherent structures,and the controversial maximum entropy production principle.
CONTENTS
I. Introduction 2A. Motivation 2B. Applications of the second law in climate research 2C. Structure of the review 3II. The second law of thermodynamics applied to theclimate system 4A. Fluxes of entropy into and out of the climate system 41. The planetary climate system 52. The material climate system 73. The transfer climate system 9B. The climate system as a heat engine 101. Heat engines and irreversibility 102. Carnot efficiency of the climate system 103. Mechanical efficiency of the climate system 11III. Irreversible processes in the climate system 12A. Derivation of the material entropy budget 121. Single-component fluids 132. Multi-component fluids 13B. Thermodynamics of a moist atmosphere 151. Irreversible phase changes 152. Diffusion of water vapor 163. Irreversible sedimentation of precipitation 16 ∗ [email protected] 4. The entropy budget of a moist atmosphere 175. The role of latent heating 18C. Irreversible entropy production in the ocean 19IV. The entropy budget of atmospheric convection 20A. Radiative-convective equilibrium (RCE) 201. Dry radiative-convective equilibrium 212. Moist radiative-convective equilibrium 22B. Theories of moist convection 231. The moist convective heat engine 232. Moist irreversible processes in radiative-convectiveequilibrium 243. The role of mixing and microphysics 26C. Convective organization and the mechanicalefficiency of moist convection 26V. The thermodynamics of tropical cyclones 27A. Potential intensity theory 28B. Work and entropy budgets of TCs 30C. Open TCs 32VI. The global circulation of the atmosphere 33A. The material entropy budget of the globalatmosphere 33B. The global atmospheric heat engine 341. A thermodynamic perspective of the globalatmospheric circulation 342. Theories for the global atmospheric heat engine 353. The atmospheric heat engine under climatechange 36C. Heat engines on other planets 37 a r X i v : . [ phy s i c s . a o - ph ] F e b
1. Rocky planets 382. Giant planets 39VII. Modeling the second law of thermodynamics 40A. Cloud-resolving models 40B. Global climate models 41VIII. Variational approaches for climate and geophysical flows 42A. Entropic energies 421. Available potential energy 432. Exergetics 44B. Statistical mechanical approaches for steady flows 461. Theoretical development 472. Applications 49C. Maximum entropy production principle controversy 51IX. Conclusions & perspectives 54References 55
I. INTRODUCTIONA. Motivation
The Earth is a highly irreversible thermodynamic sys-tem. It receives energy and entropy from the sun andradiates energy and entropy to outer space. But whilethe incoming and outgoing fluxes of energy are roughlyin balance, the Earth exports vastly more entropy thanit receives (Peixoto et al., 1991; Stephens and O’Brien,1993). For a climate that does not vary in time, thesecond law of thermodynamics requires that this net ex-port of entropy be balanced by irreversible productionof entropy within the climate system. The second lawtherefore provides a fundamental steady-state constrainton the climate system, relating a measure of its internalactivity, the irreversible production of entropy, to fluxesof entropy at its boundaries.In fact, the Earth’s climate is not steady; it has un-dergone vast changes over Earth’s history, from the icycold of Snowball Earth episodes (Hoffman et al., 1998)to the extreme warmth of the Late Cretaceous that al-lowed crocodile-like reptiles to roam the Arctic (Tardunoet al., 1998). Such climate variability occurs at a rangeof timescales (Ghil and Lucarini, 2020) and implies im-balances in the planetary energy and entropy budgets.In the context of current climate change, these energyimbalances are small relative to the total incoming andoutgoing fluxes (Trenberth et al., 2014), and the steady-state assumption provides a useful framework for under-standing the second law as applied to the climate system.A range of processes are involved in the irreversibleproduction of entropy within the climate system includ-ing the absorption and emission of radiation, the dissipa-tion of winds and ocean currents, and the movement ofwater through the hydrological cycle. Indeed, life itself isan irreversible process, although we will not discuss thebiotic generation of entropy in this review. But despitetheir ubiquity, irreversible processes are often treated in simplified ways in studies of the large-scale atmosphericcirculation (Emanuel, 2001), and numerical models of theclimate system often neglect certain irreversible processesaltogether (e.g., Pascale et al., 2011; Pauluis and Held,2002a). Moreover, interaction between the communitiesof climate scientists and physicists developing tools forthe understanding of irreversible processes remains lim-ited. Fostering collaborations between these communi-ties has the potential to reveal new methods for analyz-ing and understanding the climate system, particularlyin the context of the important challenge of predictingfuture global climate change (Lucarini et al., 2010a).The purpose of this review is to provide an introduc-tion to the application of the second law of thermody-namics to the climate system suitable both for scientistsactive within climate research and for a more generalaudience of physicists. The research frontier in climatephysics is rich with fascinating, complex problems. Atraining in traditional physics is excellent preparation forclimate research, and researchers with expertise in areasincluding statistical mechanics, fluid dynamics and phys-ical chemistry have much to offer as part of a vibrant,interdisciplinary climate science community (Marston,2011; Wettlaufer, 2016).
B. Applications of the second law in climate research
The bulk of Earth’s irreversible entropy productionoccurs as a result of radiative processes (Stephens andO’Brien, 1993). Because the irreversibility of radiativeabsorption and emission is not directly relevant to theatmospheric and oceanic circulation (Goody, 2000), ap-plications of the second law in climate research often fo-cus on a subset of the climate system that includes onlymatter and considers radiation as part of the system’ssurroundings. This perspective allows for the definitionof the material entropy budget. The steady-state mate-rial entropy budget requires a balance between materialentropy production and the net sink of entropy owingto radiative heating at high temperature and radiativecooling at low temperature (Pauluis and Held, 2002a).A number of studies have used the material entropybudget to provide theoretical constraints on the behav-ior of atmospheric circulations of various scales, includingconvective clouds (Renn´o and Ingersoll, 1996; Robe andEmanuel, 1996), tropical cyclones (Emanuel, 1986), andthe global circulation itself (Barry et al., 2002). Suchstudies generally treat the climate system as a heat en-gine, ingesting heat in a warm region, transporting itto a cool region where it is expelled, and performing anamount of work in the process. But unlike a traditionalheat engine, the work performed by the climate systemmust be dissipated within the system itself.A major challenge for heat-engine based theories ap-plied to the atmosphere is that they must properly ac-count for the influence of “moist” processes—processesassociated with water in the atmosphere. Moist processesare responsible for the bulk of the irreversible materialentropy production in Earth’s atmosphere, and this lim-its the efficiency with which the climate system’s heatengine may generate kinetic energy in winds and oceancurrents (Pauluis and Held, 2002a; Pauluis et al., 2000;Romps, 2008). The effects of irreversible moist processesare particularly relevant in the context of global climatechange (Lalibert´e et al., 2015; Singh and O’Gorman,2016), as the concentration of water vapor in the atmo-sphere is expected to increase with warming roughly fol-lowing the Clausius-Clapeyron relation (O’Gorman andMuller, 2010).The use of variational methods represents an alterna-tive approach to the application of the second law toclimate research. Maximization of entropy, under appro-priate constraints, allows for the determination of thethermodynamic equilibrium state of an isolated systemwithout solving for its evolution. The climate systemis neither isolated nor close to equilibrium, but varia-tional approaches have nonetheless been used to quantifythe ability of the climate system to perform work (e.g.,Bannon, 2005; Tailleux, 2013). Additionally, the max-imization of an entropy variable in the context of two-dimensional ideal fluid mechanics has provided a range ofinsights into geophysical flows on Earth and other plan-ets (e.g., Bouchet and Venaille, 2012; Majda and Wang,2006).An extension of the entropy maximization formalism tonon-equlibrium systems was proposed by Paltridge (1975,1978) who suggested that the climate system evolves to astate that maximizes its entropy production rate . If gen-erally applicable to the climate, this maximum entropyproduction (MEP) principle would allow for the appli-cation of variational approaches to solve for the steady-state climate directly. We emphasize, however, that theMEP principle is not implied directly by the second law,and despite some attempts to derive it from more generalprinciples (e.g., Dewar, 2005), it still lacks a solid theo-retical foundation (Bruers, 2007; Grinstein and Linsker,2007). Moreover, a number of authors have presentedmodeling results (Chang, 2019; Pascale et al., 2012b)and observational analyses (Goody, 2007) that violatethe MEP principle, at least as originally formulated. Wetherefore focus this review on more traditional applica-tions of the second law to climate research, although wepresent a summary of MEP-related research in sectionVIII.C.
C. Structure of the review
The bulk of this review is focused on Earth’s at-mosphere, where most irreversible entropy productionwithin the (material) climate system occurs. While this review primarily adopts a view of the second law focusedon entropy production, irreversibility in the climate sys-tem may also be framed in energetic terms through theconcepts of exergy (e.g., Bannon, 2005), and availablepotential energy (Lorenz, 1955). We briefly discuss theseapproaches in section VIII.A; the reader is referred toTailleux (2013) for a more complete treatment. Finally,we emphasize that this review covers only a small frac-tion of the broader research field of climate dynamics;a thorough review of the physics of climate change hasbeen recently published in this journal by Ghil and Lu-carini (2020). The remainder of the review is structuredas follows.Section II introduces the basic thermodynamic proper-ties of the climate system. We discuss methods of defin-ing the boundaries of the system, including the planetaryand material definitions used most commonly in the lit-erature. We also describe the climate system as a heatengine, and we show how classical engineering conceptssuch as the work performed and the efficiency may bemeaningfully applied to the climate system.Section III sketches a derivation of the entropy budgetof the climate system. We focus on the material entropybudget of the atmosphere, and we describe the physicaland mathematical origins of the main irreversible pro-cesses. We also briefly discuss the oceanic entropy bud-get and recent work estimating irreversible processes inthe ocean.Sections IV and V review applications of the secondlaw of thermodynamics to atmospheric convection andtropical cyclones, respectively. In particular, we highlighthow the irreversibility of moist processes fundamentallychanges the fluid dynamics of the atmosphere.Section VI considers the global atmospheric circulationfrom a thermodynamic perspective. We consider theoriesof the global atmospheric heat engine and we discuss howit may change under climate change. We also reviewresearch describing the heat engines of other planets andbodies in the Solar System and beyond.Section VII discusses some of the challenges faced indeveloping numerical models of the climate system thataccurately represent the second law of thermodynam-ics. We describe practical and theoretical limitations ofpresent modeling frameworks, and we suggest strategiesto aid future model development.Section VIII provides an introduction to variationalapproaches to understanding geophysical fluid dynamicsand the climate generally. We discuss the applicationof such approaches to atmospheric energetics and tur-bulence in large-scale geophysical flows. Here, both theclassical thermodynamic definition of entropy, as well asthe Boltzmann entropy of statistical mechanics, are em-ployed. We also discuss the controversial maximum en-tropy production (MEP) principle, which has motivatedmuch research into the climate system’s entropy budget.Section IX concludes this review with a summary anddiscussion of outstanding research questions. We partic-ularly highlight those areas that are likely to benefit fromengagement with a broader community of physicists.
II. THE SECOND LAW OF THERMODYNAMICSAPPLIED TO THE CLIMATE SYSTEM
The second law of thermodynamics states that the en-tropy S of an isolated system must increase with time(de Groot and Mazur, 1984): dSdt ≥ . (isolated system) (1)The entropy is a function of the state of the system. Itmay be defined using statistical mechanics as a measureof the number of microstates corresponding to a givenmacrostate, or in classical thermodynamics by the rela-tionship dSdt = ˙ Q rev T , (closed, reversible system) (2)valid for a closed, reversible system. Here ˙ Q rev representsa reversible heat transport from the surroundings to thesystem and T is the temperature at which this heat istransported (e.g., de Groot and Mazur, 1984; Iribarneand Godson, 1981).The climate system exchanges energy with space in theform of radiation, and it is therefore not isolated. Thesecond law for a non-isolated system may be written inthe more general form, dSdt = ˙ S e + ˙ S i , (3)where ˙ S e is the net import of entropy from the surround-ings and ˙ S i is the production of entropy within the sys-tem owing to irreversible processes (de Groot and Mazur,1984). The second law of thermodynamics requires that˙ S i ≥ dS/dt ≈
0, andthe second law of thermodynamics as applied to the cli-mate system may be written,˙ S i = − ˙ S e . (4) That is, the irreversible entropy production rate of theclimate system is equal to the net rate of export of en-tropy to space. For applications to the Earth, it willprove useful to measure entropy exchanges per unit areaof the Earth’s surface, giving ˙ S i the units of W m − K − .The steady-state entropy budget (4) states that, inorder to maintain entropy producing processes such asthose associated with winds, ocean currents, and the hy-drological cycle, the climate system must export a greaterquantity of entropy than it receives. This is manifestin the relatively high entropy contained in the radiationemitted from Earth to space compared to the lower en-tropy of the solar beam. More generally, the entropy bud-get places a fundamental constraint on the climate sys-tem by relating a measure of its internal activity, the totalirreversible entropy production, to fluxes at its bound-aries. One of the main purposes of studies of the cli-mate’s entropy budget is to leverage this constraint tobetter understand aspects of the climate system’s behav-ior.In the remainder of this section, we describe differentmethods of evaluating the entropy fluxes into and outof the climate system depending on how the system’sboundaries are defined (section II.A). We also introducethe concept of the climate system as a heat engine, andwe define the work done by the climate system and itsthermodynamic and mechanical efficiency (section II.B). A. Fluxes of entropy into and out of the climate system
Evaluating the net import of entropy into the climatesystem ˙ S e requires a proper definition of the climatesystem and its boundaries; where does the Earth’s cli-mate system end and “the surroundings” begin? Ban-non (2015) summarizes a number of possible definitionsof the climate system, but here we limit our discussion tothree common definitions used in studies of the Earth’sentropy budget (Fig. 1):1. The planetary climate system: the Earth and its at-mosphere is treated as a control volume, and the cli-mate system is defined as all substances, both mat-ter and radiation, within this volume (e.g., Bannon,2015).2.
The material climate system: the climate system isdefined to include only matter within the Earth andatmosphere, and all photons are considered part ofthe surroundings (e.g., Goody, 2000).3.
The transfer climate system: discussed in Bannon(2015), and recently advocated for by Gibbins andHaigh (2020), the transfer climate system is definedto include matter plus internal radiative transfer(photons that are emitted and absorbed by matter
TOAsurface
FIG. 1 Schematic of different definitions of the climate sys-tem and its boundary in the application of the second lawof thermodynamics. The planetary system is defined as acontrol volume including all particles (matter and radiation)encompassed by a fictitious surface denoted “top of the at-mosphere” (TOA). The material system includes only matter(circles) and excludes all radiation (arrows) as part of the sur-roundings. The transfer system (Gibbins and Haigh, 2020)includes internal radiative transfer (photons that are emittedand absorbed by matter within the system; solid black ar-rows), but excludes external radiative transfer (photons thatare incident from the sun or emitted directly to outer space;dashed black arrows). within the system) but to exclude external radia-tive transfer (photons that are incident from thesun or emitted directly to space).Although each perspective provides a consistent de-scription of the climate system, the magnitude of theentropy import ˙ S e and the implied irreversible entropyproduction ˙ S i differs greatly between the planetary, ma-terial, and transfer definitions, and previous authors havedisagreed on which perspective is most relevant for stud-ies of the climate (e.g., Essex, 1984, 1987; Goody, 2000).Following Goody (2000) and a number of other authors(e.g. Lucarini et al., 2010a; Ozawa et al., 2003; Pascaleet al., 2011), we will argue that the material climate sys-tem is most relevant to understanding the dynamics ofthe atmosphere and ocean, and the material entropy bud-get will be the focus of much of the later sections of thisreview.
1. The planetary climate system
The planetary climate system consists of a control vol-ume bounded by a fictitious surface beyond the atmo-sphere which we will refer to as the “top of the atmo-sphere” (TOA; Fig. 1). Fluxes of entropy through theTOA are carried by photons emitted by the sun (short-wave radiation) and those emitted by the Earth and at-mosphere (longwave radiation). Defining Ω as the vol-ume of the climate system, we may write the net import of entropy into the system as,˙ S e = − A (cid:90) ∂ Ω ( J SW + J LW ) d A, where ∂ Ω represents the boundary of Ω, in this case theTOA, A is the surface area of Earth, J is the radiantflux of entropy out of the climate system, and subscripts SW and LW refer to shortwave and longwave radiation,respectively. For a system approximately in steady state,we must also have a balance between the shortwave andlongwave radiant energy fluxes F ,1 A (cid:90) ∂ Ω ( F SW + F LW ) d A = 0 . Previous authors have estimated the entropy fluxes J LW and J SW from both observations (Kato and Rose,2020; Stephens and O’Brien, 1993) and climate models(Li et al., 1994; Pascale et al., 2011). Before we discussthese estimates, however, it is useful to consider the plan-etary entropy budget for a simplified model of the climatesystem in order to build some intuition of the magnitudeand behavior of various components of ˙ S e .The simple model is described schematically in Fig.2; it is similar to models presented in Bannon (2015)and Kato and Rose (2020), and our discussion of the dif-ferent entropy production rates follows that of Gibbinsand Haigh (2020). The model is horizontally homoge-neous, representing globally-averaged conditions, and itconsists of a surface and a single-layer atmosphere. Boththe surface and atmosphere are assumed to be completelyopaque to longwave radiation and to behave as blackbod-ies for radiation in the longwave portion of the electro-magnetic spectrum . The atmosphere is assumed to betransparent to shortwave radiation, and the surface hasa fixed shortwave albedo of α , reflecting a fraction α ofthe incoming solar radiation to space and absorbing therest. Energy and entropy transports in this model oc-cur via radiative fluxes between the surface, atmosphere,and space, and via turbulent fluxes of latent and sensibleheat between the surface and the atmosphere.Assuming steady state, we may write energy balanceequations for the TOA, atmosphere, and surface, respec- We follow the convention in atmospheric science to refer to a fluxas the transport of a quantity per unit area (also known as fluxdensity). It is sometimes suggested that the greenhouse effect is incom-patible with the second law of thermodynamics. The one-layermodel shows this is not true; the model’s surface is maintainedat a temperature higher than the effective emission temperatureof the planet by the greenhouse effect associated with the ab-sorption of longwave radiation, and as shown below, the modelsatisfies the second law. Latent heat refers to the energy embodied in water vapor that isreleased on condensation. 𝑇 " = 255 K𝑇 ( = 288 K 𝐹 +, + 𝐹 ., 𝐹 ./↑ = 𝛼𝛿 𝜎𝑇 𝐽 ./↑ = 𝜒(𝑢)𝛿 𝜎𝑇 𝐹 +/ > = 𝜎𝑇 (7 𝐽 +/ > = 𝜎𝑇 (9 𝐹 +/ ? = 𝜎𝑇 "7 𝐽 +/ ? = 𝜎𝑇 "9 𝐹 @A ↓ = 𝛿 𝜎𝑇 𝐽 @A ↓ = 𝛿 𝜎𝑇 FIG. 2 A simple model for Earth’s climate and the associated vertical energy and entropy fluxes. The climate system isassumed to be horizontally homogeneous, and it includes a surface, with temperature T s , and a single-layer atmosphere, withtemperature T a . Arrows show energy fluxes F and their corresponding entropy fluxes J expressed per unit area. The subscripts LW and SW denote shortwave and longwave radiant fluxes, respectively, and the subscripts s and a refer to the surface andatmosphere, respectively. Shortwave radiation is divided into its upward and downward components. The turbulent flux ofenergy from the surface to the atmosphere is made up of a sensible heat flux F SH and a latent heat flux F LH . Other terms inthe equations are described in the text. tively, given by Kato and Rose (2020):(1 − α ) F ↓ SW = F aLW , (5a) F sLW + F LH + F SH = 2 F aLW , (5b)(1 − α ) F ↓ SW = F sLW + F LH + F SH , (5c)where F ↓ S is the downward solar energy flux input to theEarth, F sL and F aL are the longwave energy fluxes from thesurface and atmosphere, respectively, and F LH and F SH are the latent and sensible heat fluxes from the surfaceto the atmosphere, respectively (Fig. 2).The longwave energy fluxes from the surface and at-mosphere are given by the well-known Stefan-Boltzmannlaw F = σT , (6)where F is the energy flux, T is the temperature of theemitting body and σ is the Stefan-Boltzmann constant.Approximating the sun as a blackbody, the downwardsolar energy flux at the TOA is given by, F ↓ SW = δ sun σT , where δ sun = Ω cos φπ , with Ω = 6 . × − being thesolid angle subtended by the sun’s disk (Stephens andO’Brien, 1993) and φ the zenith angle of the sun’s rays.Here we take a global mean value of cos φ = 0 . T sun = 5777 K (Peixoto et al.,1991), and we set α = 0 . T s . We chooseto set the surface temperature to roughly match Earth’sglobal-mean surface temperature T s = 288 K, noting thatthe purpose of this model is to aid understanding of the entropy budget rather than to predict planetary temper-atures. Using the energy balance equations (5) and theparameters given above, this constrains the atmospherictemperature, which for this model is equal to the effec-tive emission temperature of the planet, to be T a = 255K.We now evaluate the planetary entropy budget for theone-layer model. The entropy fluxes associated with radi-ation of a given wavelength and angular distribution maybe derived from the fundamental statistical mechanics ofa Boson gas (Rosen, 1954), or through a number of semi-classical methods (Ore, 1955; Wu and Liu, 2010). For ablackbody, a formula for the entropy flux J may be de-rived by integrating the spectral entropy flux distributionover all frequencies to give (Wu and Liu, 2010), J = 43 σT . (7)Combining (7) and (6), the blackbody entropy flux maybe expressed in terms of the energy flux F as, J = 43 FT . (8)The entropy flux emitted by a blackbody is larger, by afactor of 4 /
3, than the entropy loss of the emitting object
F/T . This additional entropy transport may be inter-preted as the irreversible entropy production associatedwith emitting radiation into a vacuum (Feistel, 2011) . An elegant derivation of (8) is presented by Feistel (2011). Con-sider two parallel plates held at fixed temperatures exchangingenergy through radiation. The energy flux from each plate maybe described by the Stefan-Boltzmann law (6), and it will pro-duce a transfer of heat from the hotter plate to the colder plate.By the second law of thermodynamics, this energy exchange must
Eq. (7) may be used to evaluate the entropy fluxes fromthe surface and atmosphere in the one-layer model.Due to the irreversibility associated with reflection, theupward and downward shortwave entropy fluxes must betreated separately. The downward shortwave flux of en-tropy is given simply by, J ↓ SW = 43 σδ sun T , (9)representing the sun’s blackbody entropy flux reducedby the the factor δ sun . The same approximation cannotbe used for the upward flux because of the change of itsangular distribution upon reflection. Instead, we followStephens and O’Brien (1993) and apply an approximateformula for the entropy flux of the reflected beam underthe assumption of Lambertian reflection given by, J ↑ SW = 43 σT u ( a + a ln( u )) , (10)where u = αδ sun , and a = 0 . a = − . S i = 1279 mW m − K − .Despite the simplicity of the one-layer model, its en-tropy budget is similar to more detailed estimates ofEarth’s planetary entropy budget based on observations(table I). For example, Stephens and O’Brien (1993) usedsatellite observations of TOA radiation to estimate theplanetary entropy budget, finding a similar dominanceof the longwave fluxes and a value of the total entropy be associated with positive irreversible entropy production, withthe rate of irreversible entropy production tending to zero as thetemperature difference between the plates approaches zero. Feis-tel (2011) showed that the only functional form for the entropyflux associated with the radiation from each plate consistent withthis expectation is that described by (8). import ˙ S e roughly 2% smaller in magnitude than thatof the one-layer model. This difference is partially ac-counted for by the one-layer model’s neglect of temper-ature variations within the atmosphere. Lesins (1990)showed that the outgoing longwave entropy flux is maxi-mized for an isothermal atmosphere, with meridional andvertical temperature variations reducing the flux by a fac-tor of the order of 1%.Kato and Rose (2020) also provided estimates of theentropy flux at the TOA based on satellite observations.They applied the simple blackbody formula (8) to bothincoming and outgoing shortwave radiation. This ne-glects the irreversible entropy production associated withdiffuse reflection, resulting in an underestimate of the en-tropy flux by reflected solar radiation. But the overallplanetary entropy budget is nevertheless broadly similarto that of the one-layer atmosphere model.On the other hand, the observational study of Peixotoet al. (1991) and a number of studies based on global cli-mate models (Fraedrich and Lunkeit, 2008; Pascale et al.,2011) present planetary entropy budgets that are incon-sistent with the one-layer model, with estimates of theentropy import ˙ S e J (cid:48) = FT , (11)where F is the radiative energy flux and T is the tem-perature of the emitting object. This definition is ap-pealing, because it gives the radiant entropy flux as be-ing equal to the loss of entropy by the emitting ob-ject, but it fails to account for the irreversible natureof spontaneous emission and absorption represented bythe 4 /
2. The material climate system
To motivate study of the material entropy budget, con-sider a thought experiment in which the heating and cool-ing owing to radiative absorption and emission within theclimate system is replaced by identical heating and cool-ing rates produced by a reversible mechanism. Such achange would have no effect on the matter within the cli-mate system; the equations governing the fluid dynamics
Model J TOA SW J TOA LW − ˙ S e − ˙ S trans e − ˙ S mat e one-layer model (Fig. 2) 31 1247 1279 109 39Kato and Rose (2020) obs. -55 1238 1183 76 49Stephens and O’Brien (1993) obs. 20 1230 1250Peixoto et al. (1991) obs. -41 a a a b a
101 52Fraedrich and Lunkeit (2008) PlanetSim 880 a
69 35Goody (2000) GISS 52TABLE I Observational (obs.) and model-based estimates of the top of atmosphere entropy fluxes of shortwave J TOA SW andlongwave J TOA LW radiation and the planetary − ˙ S e , transfer − ˙ S trans e and material − ˙ S mat e entropy export rates of the climatesystem (per unit area of Earth’s surface) from various prior studies and for the one-layer model summarized in Fig. 2. Fluxestoward outer space are defined as positive, and units are mW m − K − . a Entropy fluxes estimated using (11). b Mean over 7 models participating in the sixth phase of the Coupled Model Intercomparison Project (CMIP6) (Eyring et al., 2016). of the atmosphere and oceans are only concerned withradiation insofar as it heats or cools the fluid (Goody,2000; Pascale et al., 2011). The atmospheric circulation,hydrological cycle, and ocean currents would behave ex-actly as before. For this reason, Goody (2000) advocatedfor a view of the entropy budget that focuses exclusivelyon matter rather than radiation.The material climate system includes all matter withinthe Earth, atmosphere, and oceans but considers radia-tion as part of the surroundings (Bannon, 2015; Goody,2000). The import of entropy into the material system˙ S mat e occurs through the heating and cooling of matterthat absorbs and emits radiation, and it may be written(Goody, 2000), ˙ S mat e = 1 A (cid:90) Ω ˙ q rad T ρdV, (12)where ˙ q rad is the net radiative heating rate per unitmass , ρ is the density, and the integral is over Ω, rep-resenting the entire climate system. We may also de-fine the irreversible material entropy production ˙ S mat i ,which by the steady-state material entropy budget sat-isfies ˙ S mat i = − ˙ S mat e . The steady-state material entropybudget may then be written,˙ S mat i = − A (cid:90) Ω ˙ q rad T ρdV. (13)Since the material entropy budget incorporates the ef-fect of radiation as an external (and reversible) heating, Here, and throughout this review, we use lower-case letters torefer to intensive quantities expressed per unit mass and upper-case letters to refer to quantities defined by integrals over theentire system of interest and expressed per unit surface area. the material irreversible entropy production rate ˙ S mat i ac-counts for only a portion of the total irreversible entropyproduction of the climate system ˙ S i , while the remainder˙ S rad i is associated with irreversible entropy productionowing to radiation. As we will show in detail in sectionIII, the material entropy production includes irreversibleprocesses such as frictional dissipation, molecular heatdiffusion, irreversible mixing, and irreversible chemicalreactions.As a starting point, we consider the material entropybudget of the simple one-layer model introduced in theprevious subsection. In this model, radiation acts tocool the atmosphere and heat the surface, with turbu-lent fluxes transporting energy from the surface to theatmosphere to balance. The net radiative heating of theatmosphere ˙ Q rad may be evaluated based on the radiativeenergy fluxes shown in Fig. 2,˙ Q rad = σT s − σT a . (14)Using the energy balance equations for the one-layermodel (5), one may show that the net radiative heatingof the surface is given by − ˙ Q rad , and the steady-statematerial entropy budget of the one-layer model may beexpressed, ˙ S mat i = − ˙ Q rad (cid:18) T a − T s (cid:19) . Since T s > T a and ˙ Q rad is negative, the entropy pro-duction ˙ S mat i is positive, as required by the second law.The one-layer model is heated by radiation at the surfaceat temperature T s and cooled in the atmosphere at thelower temperature T a . This produces a net entropy sinkwhich in steady state must be balanced by the materialentropy production ˙ S mat i .For the parameters chosen, the one-layer model givesa material entropy production rate of ˙ S mat i = 39 mWm − K − . Given the assumptions of the model, this is arather crude estimate of the material entropy productionin the climate system, but it highlights the small magni-tude of ˙ S mat i compared to the total irreversible produc-tion rate ˙ S i , implying that the bulk of the irreversibleentropy production in the climate system occurs due toradiative processes (Essex, 1984, 1987; Goody, 2000; Liet al., 1994; Stephens and O’Brien, 1993).More detailed estimates of the material entropy bud-get of Earth’s climate system confirm the picture above(table I). Studies based on observations (Kato and Rose,2020; Peixoto et al., 1991) as well as global climate modelsimulations (Fraedrich and Lunkeit, 2008; Goody, 2000;Lembo et al., 2019; Pascale et al., 2011) have estimatedthe material entropy export − ˙ S mat e , finding values in therelatively broad range of 35-60 mW m − K − . The largerange in such estimates is partly due to methodologicaldifferences across studies. For example, Peixoto et al.(1991) considered only global- and annual-mean radia-tive fluxes and temperature profiles in order to estimate˙ S mat e , while Kato and Rose (2020) also took into ac-count spatial and temporal variations. But differencesalso arise because the spatial distribution of radiativeheating ˙ q rad is strongly dependent on properties such asthe surface albedo and the distribution of clouds and wa-ter vapor in the atmosphere, and it is therefore difficultto estimate accurately from observations and dependenton uncertain parameterizations in models. As a result,even among studies applying similar methodologies toclimate model output (Fraedrich and Lunkeit, 2008; Pas-cale et al., 2011), the estimated value of ˙ S mat e can varysubstantially.Under steady-state conditions, estimation of the en-tropy import leads directly to an estimate of the materialentropy production of the climate system. However, dif-ferences between estimates of − ˙ S mat e and ˙ S mat i of up to30% have been reported in the literature (Lembo et al.,2019). In principle, such differences may result from im-balances in the entropy budget due to climate variation(Goody, 2000), but estimates of the imbalance in theplanetary energy balance (Trenberth et al., 2014) sug-gest that such differences are likely to be on the order of afew mW m − K − . Rather, differences between estimatesof the entropy import and estimates of material entropyproduction reflect the difficulty of diagnosing irreversibleprocesses using available observations or using standardmodel outputs. We further discuss the issues surround-ing the estimation of irreversible processes in the climatesystem in section VI.A and in climate models in sectionVII.
3. The transfer climate system
While the majority of studies of Earth’s entropy bud-get adopt one of the two definitions discussed above, Gibbins and Haigh (2020) have recently advocated foran alternate definition that is intermediate between theplanetary and material perspectives which they refer toas the transfer climate system (also discussed in Bannon,2015, as their case “MS2”). The transfer climate systemis similar to the material climate system, but it addi-tionally includes radiation that is “internal” to the cli-mate system (Fig. 1). Internal radiation corresponds tophotons that transport energy between different materialelements of the climate system, in contrast to those pho-tons that are incident on the Earth from the sun or thatare emitted to outer space. Since the transfer approachincludes some, but not all, radiation as part of the cli-mate system, it gives an irreversible entropy productionrate ˙ S trans i whose magnitude is between the planetaryand material values.Gibbins and Haigh (2020) argue that the transfer cli-mate system provides an entropy budget that is morerobust to details of internal heat transport mechanismswithin the climate system. In particular, they note that,from the perspective of the transfer entropy budget, heattransport from high to low temperature is associatedwith the same amount of irreversible entropy productionwhether it is caused by radiative fluxes or conductivefluxes. But only in the latter case would this entropyproduction be included in the material entropy budget.For the transfer climate system, the entropy importrate ˙ S trans e is equal to the sum of the entropy tenden-cies associated with the absorption of solar radiation andthe emission of longwave radiation directly to space. Ac-cording to the one-layer model, the atmosphere emitsan amount of radiation σT a directly to space, while thesurface absorbs an equal amount of radiation from thesun. Assuming steady-state conditions, we may there-fore write the irreversible entropy production rate ˙ S trans i for the one-layer model as˙ S trans i = σT a (cid:18) T a − T s (cid:19) . This may be evaluated with the parameters of the modelto give ˙ S trans i = 109 mW m − K − .A disadvantage of using the transfer approach is thatthe transfer entropy import rate ˙ S trans e depends on theorigin and destination of each photon that enters the cli-mate system, rather than just the net radiative heat-ing rate or the TOA fluxes, making its estimation fromobservations and models more involved. Nevertheless,Kato and Rose (2020) have recently estimated the trans-fer entropy budget from observations, and some of itscharacteristics may be deduced from the results of pre-vious studies using global climate models (Fraedrich andLunkeit, 2008; Pascale et al., 2011). As expected, themagnitude of the entropy export − ˙ S trans e is betweenthe corresponding values for the planetary and mate-rial entropy budgets (table I). Compared to the obser-vational and climate-model based estimates, the simple0one-layer model overestimates the transfer entropy pro-duction rate. This is likely because of its neglect of solarabsorption in the atmosphere, which leads to an artifi-cially high value of the solar absorption temperature (cf.Gibbins and Haigh, 2020).But even detailed model-based estimates of the trans-fer entropy production rate differ from each other con-siderably. This highlights our limited knowledge of thetransfer entropy production rate, which has only recentlybeen explicitly defined in the context of the climate sys-tem’s entropy budget (Bannon, 2015; Gibbins and Haigh,2020). Better quantification of the transfer entropy bud-get and further understanding of its relationship to thethermodynamics of the climate system present promisingavenues for future research. B. The climate system as a heat engine
The climate system is often described as a heat engine,transporting energy from the warm tropical surface to thecold polar troposphere and producing atmospheric andoceanic circulations in the process (e.g., Bannon, 2015;Barry et al., 2002; Brunt, 1926; Lalibert´e et al., 2015;Lorenz, 1967; Pauluis, 2011). But there are some im-portant differences between the climate system and theclassic engineering account of a heat engine. In the fol-lowing, we clarify how concepts such as the work outputand the thermodynamic efficiency of a heat engine maybe sensibly applied to the climate system.
1. Heat engines and irreversibility
Consider a heat engine operating between two thermalreservoirs at different temperatures. The engine ingestsheat at a rate ˙ Q in from the warm reservoir at a temper-ature T in , transporting it to the cool reservoir at tem-perature T out where it is expelled at a rate ˙ Q out . In theprocess, the engine is able to perform work at the rate˙ W ext . Here we include the subscript ext to emphasizethat this work is done on an external body. For instance,the engine may be used to drive a piston that acceleratesa locomotive. The eventual dissipation of the locomo-tive’s kinetic energy occurs outside of the engine.The action of a heat engine may be described by com-bining the first and second laws of thermodynamics un-der steady-state conditions to form the Gouy-Stoudolatheorem (e.g., Bannon, 2015), η C ˙ Q in = ˙ W ext + ˙ S mat i T out , (15)where ˙ S mat i represents the irreversible entropy produc-tion rate of the engine and η C = T in − T out T in . If the engine is perfectly reversible, it produces work withan efficiency given by η C , equal to the Carnot efficiency.Irreversible processes decrease the work output relativeto this theoretical maximum. In the engineering context,irreversible entropy production is considered “lost work”,and the aim of the engineer is to reduce ˙ S mat i as much aspossible.
2. Carnot efficiency of the climate system
The climate system does not have a warm or cold reser-voir, and the heating rates ˙ Q in and ˙ Q out and the associ-ated temperatures are more difficult to define. Moreover,the climate system as a whole cannot perform work onany external body. For the climate system, we there-fore have that ˙ W ext = 0 and any traditionally definedefficiency is also zero. Nevertheless, previous authorshave defined an efficiency of the climate system in variousways.For example, Bannon (2015) considers ˙ S i T out to be ameasure of the activity of the atmosphere and oceans.The author defines the efficiency of the climate systemas the ratio of this activity to the heat input ˙ Q in . Since˙ W ext = 0, (15) then gives that η C = S mat i T out Q in . ( ˙ W ext = 0)That is, the efficiency of the climate system is equal tothe Carnot efficiency. The task is then to determine theeffective input and output temperatures and heating rate˙ Q in . Since there is no warm or cold reservoir, the heatingrates must be defined in an averaged sense. In particular,we may take the energy input as the sum over all regionsthat experience net heating (e.g., Bannon, 2015),˙ Q in = 1 A (cid:90) Ω ˙ q +rad ρdV, (16)where ˙ q +rad is the net radiative heating rate when it ispositive and zero otherwise. Similarly, we may define theeffective temperature of heat input by,˙ Q in T in = 1 A (cid:90) Ω ˙ q +rad T ρdV.
Making similar definitions of the heat output and outputtemperature based on the radiative cooling rate, (15) maybe written ˙ S mat i = ˙ Q in (cid:18) T out − T in (cid:19) . (17)Here we have assumed that ˙ W ext = 0, and hence that ˙ Q in = ˙ Q out for the climate system. Comparing with (13), theheat engine relation above may be seen to be equivalentto the steady-state material entropy budget.1For the simple one-layer atmosphere model describedby Fig. 2, the heat input ˙ Q in is given by ˙ Q rad in (14), andthe input and output temperatures are those of the sur-face and the atmosphere, respectively. The model there-fore has a Carnot efficiency of η C = 11%. Bannon (2015)found a slightly lower value of η C = 8% using a simi-lar single-layer model of the climate system which allowsfor atmospheric absorption of shortwave radiation, whileGibbins and Haigh (2020) found Carnot efficiencies inthe range 6-12% also using highly simplified models ofthe climate system.It is important to note that the heating rate ˙ Q in is notan external parameter for a given planet (as it would bein traditional heat engine analysis), but it depends onfeatures such as the surface albedo and the cloud andwater vapor distribution, which may vary with climate.Furthermore, the definitions given above for the inputand output heating rates are non-unique (Bannon, 2015).This non-uniqueness has some parallels in the ambigu-ity of defining the boundaries of the climate system inour discussion of planetary, material, and transfer en-tropy production rates above. For example, taking ˙ Q out as cooling by longwave emission to space and Q in as so-lar absorption gives an efficiency based on the transferclimate system (Gibbins and Haigh, 2020). This per-spective was used in Bannon and Lee (2017) to deriveapproximate upper bounds to the Carnot efficiencies ofEarth, Mars, Venus, and Titan (see also section VI.C).
3. Mechanical efficiency of the climate system
While the climate system cannot perform work on anexternal body, the atmosphere and ocean perform workon themselves and each other, and this work drives thewinds and ocean currents. More specifically, work rep-resents a conversion between kinetic energy and inter-nal or potential energy within the climate system (e.g.,Lucarini, 2009; Lucarini et al., 2010a). This conversionmay occur reversibly via motions of the ocean and atmo-sphere, or irreversibly via dissipative processes. In thelatter case, kinetic energy may be transformed into inter-nal or potential energy, but the reverse transformation isprohibited by the second law of thermodynamics.In Earth’s climate system, kinetic energy dissipationoccurs via two processes:1. Frictional dissipation of the winds and ocean cur-rents that occurs as a result of the turbulent cas-cade of kinetic energy to scales small enough forviscosity to act.2. Frictional dissipation in the microscopic shearzones surrounding particles that have apprecia-ble sedimentation velocities relative to the fluid(e.g., raindrops and snowflakes) (Pauluis and Held,2002a; Pauluis et al., 2000). At steady state, the rate at which the climate systemperforms reversible work ˙ W rev must be equal to the totaldissipation rate owing to these two processes. We maytherefore write the steady-state mechanical energy bud-get of the climate system as (Pauluis and Held, 2002a;Romps, 2008), ˙ W rev = ˙ D fric + ˙ D sed , (18)where ˙ D fric is the rate of frictional dissipation of thewinds and ocean currents, and ˙ D sed is the rate of dissipa-tion associated with the sedimentation of precipitation.We may also define the rate of generation of kinetic en-ergy associated with winds and ocean currents by ˙ W K ;in steady state, this must be equal to ˙ D fric . Accordingto (18), ˙ W K accounts for only a portion of the reversiblework performed by the climate system; the remainder isused to lift water upwards through the atmosphere tobalance the downward irreversible flux of water owing toprecipitation (Pauluis and Held, 2002a).Since ˙ W K represents the work responsible for power-ing the atmospheric and oceanic circulation, it may beconsidered to be the “useful” component of the total re-versible work. This motivates the definition of the me-chanical efficiency of the climate system η M by η M = ˙ W K ˙ Q in . (19)The mechanical efficiency refers to the efficiency withwhich the climate system generates and dissipates ki-netic energy of the winds and ocean currents (Goody,2003; Pauluis and Held, 2002a). It is similar to the clas-sic concept of the efficiency of a heat engine, except thatthe useful work ˙ W K is done on, and dissipated within,the system itself.To examine the factors affecting the mechanical effi-ciency, we use the fact that the dissipation rates ˙ D fric and ˙ D sed are associated with irreversible entropy sourceswhich we denote ˙ S fric i and ˙ S sed i , respectively. This allowsthe heat-engine relation (17) to be written˙ Q in (cid:18) T out − T in (cid:19) = ˙ S fric i + ˙ S sed i + ˙ S nm i , where ˙ S nm i represents irreversible material entropy pro-duction by non-mechanical processes (Goody, 2003). Aswe will show in more detail in section III, the entropysource ˙ S fric i may be written in terms of the dissipationrate ˙ D fric and an effective temperature T fric ,˙ S fric i = ˙ D fric T fric . Since ˙ D fric = ˙ W K in steady state, we may combine theprevious two equations to give˙ W max = ˙ W K + T fric (cid:16) ˙ S sed i + ˙ S nm i (cid:17) , (20)2where ˙ W max = (cid:18) T fric T out (cid:19) η C Q in is the maximum rate at which work can be performedby the system for a given T fric , achieved when there areno other irreversible entropy sources besides that associ-ated with frictional dissipation of the winds and oceancurrents (Pauluis and Held, 2002a). Note that, since wegenerally expect T fric > T out , the mechanical efficiencyimplied by the performance of work at a rate ˙ W max ishigher than the Carnot efficiency η C . This is possiblebecause the work is being performed on the system itself,providing an additional dissipative heat source D fric . Therate of work ˙ W max therefore does not represent the max-imum work that can be done on an external body, whichis limited by Carnot’s theorem to not exceed that of anideal Carnot engine (Bister et al., 2011; Hewitt et al.,1975; Renn´o and Ingersoll, 1996). Rather, ˙ W max repre-sents the maximum rate of work that would be performedby an ideal Carnot engine in which the heat input andoutput is given by the combination of radiative heatingand cooling and dissipative heating experienced by theatmosphere (Lucarini, 2009).For a given value of ˙ W max , the mechanical efficiency ofthe climate system is determined by the amount of en-tropy produced by precipitation sedimentation and non-mechanical irreversible processes and the effective tem-perature of frictional dissipation. We shall see in thefollowing section that processes associated with water,including diffusion of water vapor and irreversible phasechange, are responsible for most of the non-mechanical ir-reversible entropy production in the atmosphere. Theseprocesses, coupled with the work required to lift waterupward through the atmosphere, reduce the mechanicalefficiency of the atmosphere relative to a hypotheticalatmosphere that does not contain water, and they exerta strong influence on the dynamics of the atmosphericcirculation. III. IRREVERSIBLE PROCESSES IN THE CLIMATESYSTEM
We now consider in detail the different processes thatcontribute to irreversible entropy production in the cli-mate system. Because of its clear relationship to workand kinetic energy generation, we focus on material en-tropy production in the atmosphere and oceans. We firstsketch the derivation of the material entropy budget forboth a single-component and multi-component fluid (sec-tion III.A). Readers familiar with the governing equationfor a fluid’s entropy (35) may proceed to the followingsections where we consider the application of these re-sults to the atmosphere (section III.B) and ocean (sectionIII.C) more specifically. A number of previous authors have provided more detailed treatments focused on theatmosphere (e.g., Gassmann and Herzog, 2015; Hauf andH¨oller, 1987; Pauluis, 2000), the ocean (e.g., Gregg, 1984)and in a more general context (e.g., de Groot and Mazur,1984).
A. Derivation of the material entropy budget
We begin with the second law of thermodynamics ap-plied to a fluid element of unit mass in the atmosphereor ocean. If the fluid’s interactions are purely reversible,we have, dsdt = ˙ q rev T , where s is the entropy of the fluid element and ˙ q rev is thereversible heating rate, both expressed per unit mass. Wealso have the first law of thermodynamics, dudt = ˙ q + ˙ w, where u is the internal energy of the fluid element, ˙ w israte of work done on the fluid element by its environment,and ˙ q is the heating rate. Assuming reversible conditions,˙ q = ˙ q rev , and the work is given by ˙ w rev = − p dαdt , where p is the pressure and α = ρ − is the specific volume of thefluid element. Combining the above equations, we have dudt = T dsdt − p dαdt . (21)This is the fundamental thermodynamic relation linkingentropy to other state variables for any substance of fixedcomposition (see e.g., de Groot and Mazur, 1984; Landauand Lifshitz, 1987).While (21) was derived for reversible conditions andunder the assumption of thermodynamic equilibrium, itmay be applied under more general circumstances pro-vided an approximation known as local thermodynamicequilibrium is valid (e.g., de Groot and Mazur, 1984).This approximation allows for thermodynamic functionssuch as temperature, pressure, and entropy, to be definedlocally within a fluid as a function of space and time. Inthe bulk of the atmosphere and ocean, local thermody-namic equilibrium is a very good approximation. Theexception is at very high altitudes ( (cid:38)
80 km), where thedensity of the gas becomes so low that molecular colli-sions become infrequent (e.g., Houghton, 2002). But thisregion accounts for a trivial fraction of the atmosphere’smass, and it is therefore reasonable to assume (21) isvalid when considering the entropy budget of the atmo-sphere or climate system as a whole (e.g., Lesins, 1990;Lucarini, 2009).3
1. Single-component fluids
We consider first a fluid made of a single substance.The equation governing the specific internal energy u ofsuch a fluid under the influence of radiation may be writ-ten (de Groot and Mazur, 1984), ∂ρu∂t + ∇ · ( ρ v u + D u ) = − p ∇ · u + ρ ˙ q rad + ρ(cid:15), (22)where ρ is the fluid density, v is the fluid velocity, D u isthe heat flux owing to molecular diffusion, and ˙ q rad and (cid:15) are the heating rates owing to radiation and frictionaldissipation, respectively, expressed per unit mass of thefluid. The first term on the right-hand side gives the rateof work done on the fluid by its surroundings.Eq. (22) is an Eulerian equation for the internal energyas a function of space and time, while (21) is a Lagrangianequation valid for a given element of fluid. These view-points may be related to one another by expressing thematerial derivative d/dt , representing the rate of changefollowing a given fluid element, in terms of its Euleriancounterpart, dudt = ∂u∂t + v · ∇ u. The internal energy equation may then be written in La-grangian form as, ρ dudt = − ρp dαdt + ρ ˙ q rad − ∇ · D u + ρ(cid:15), (23)where we have used the equation for mass continuity, ∂ρ∂t + ∇ · ( ρ v ) = 0 , (24)and we have rewritten the work term in terms of theLagrangian rate of change of specific volume α .Combining the internal energy equation (23) withthe fundamental thermodynamic relation (21) and us-ing mass continuity, we may write an explicit equationgoverning the fluid’s entropy, ∂ρs∂t + ∇· ( ρ v s )+ ∇· (cid:18) D u T (cid:19) − ρ ˙ q rad T = ρ(cid:15)T − D u · ∇ TT . (25)This represents the local Eulerian entropy budget of asingle-component fluid. Terms on the left-hand side give,from left to right, the local rate of change of entropy, theflux divergence of entropy by fluid motions, the flux di-vergence of entropy owing to molecular heat diffusion,and the entropy tendency due to radiative heating andcooling. The right-hand side contains the entropy pro-duction due to irreversible processes.The connection between the local Eulerian entropybudget and the material entropy budget of the climatesystem may be readily seen by integrating (25) in space.Since there are no advective and molecular fluxes at the top of the atmosphere, the flux divergences on the left-hand side vanish on integration over the entire climatesystem. Further considering steady-state conditions, thetime tendency also vanishes, and we have − A (cid:90) Ω ˙ q rad T ρdV = ˙ S fric i + ˙ S heat i , (26)where we have defined,˙ S fric i = 1 A (cid:90) Ω ρ ˙ s fric i dV = 1 A (cid:90) Ω ρ(cid:15)T dV, (27)˙ S heat i = 1 A (cid:90) Ω ρ ˙ s heat i dV = − A (cid:90) Ω D u · ∇ TT dV. (28)Comparing (26) to the material entropy budget (13), wefind that the material entropy production is given by˙ S mat i = ˙ S fric i + ˙ S heat i . For a planet comprised of a single-component fluid, there are two processes that lead toirreversible material entropy production; molecular heatdiffusion and viscous dissipation.It is important to note that our derivation of (25) re-quired only the internal energy equation and the funda-mental thermodynamic relation, which may be taken asthe relation defining entropy. Eq. (25) contains no ad-ditional information about the flow that is not alreadycontained in the energy budget (Romps, 2008). Rather,the additional information provided by the second law iscontained in the requirement for the irreversible entropyproduction terms on the right-hand side of (25) to bepositive (Gassmann and Herzog, 2015). This puts con-straints on the form of the molecular heat flux D u andthe viscous dissipation ρ(cid:15) .For example, it is easy to show that the second law issatisfied for the simple case of Fickian diffusion of tem-perature, for which D u = − κ ∇ T , for some κ ≥
0. Theassociated irreversible entropy production due to heatdiffusion is (Peixoto et al., 1991), ρ ˙ s heat i = κ (cid:18) |∇ T | T (cid:19) , which is positive definite as required by the second law.
2. Multi-component fluids
The budget equation (26) is valid for a fluid whose com-position is invariant in time and space. But the dynam-ics of Earth’s atmosphere and ocean are both stronglyinfluenced by their variable composition. In particular,irreversible entropy production in the atmosphere is dom-inated by processes associated with water in all its phases(Pauluis and Held, 2002a). We therefore must considerEarth’s atmosphere as a multi-component fluid when dis-cussing its entropy budget.We consider a fluid that is a mixture of N species, andwe denote the density of species x by ρ x . The continuity4equation for each species may be written ∂ρ x ∂t + ∇ · ( ρ x v + D x ) = ρ ˙ χ x . where the velocity v is the barycentric velocity, givenby the mass-weighted mean velocity over all species(de Groot and Mazur, 1984), and D x is the non-advectiveflux of species x , representing processes such as Brownianmotion of molecules and, in the atmosphere, sedimenta-tion of hydrometeors such as raindrops and snowflakes(Gassmann and Herzog, 2015).The quantity ˙ χ x represents the mass source of species x per unit mass of the mixture due to chemical reactions.Mass conservation requires that (cid:88) x ˙ χ x = 0 . For example, in the atmosphere, condensation representsa source of liquid water and a sink of water vapor of equalmagnitude. Since by definition the barycentric velocitygives the velocity of the center of mass of an elementof the fluid mixture, mass conservation also requires thetotal non-advective mass flux to sum to zero: (cid:88) x D x = 0 . (29)Combining the previous three equations, it may be shownthat density of the mixture ρ = (cid:80) x ρ x satisfies the con-tinuity equation (24).It is useful to define the mass fraction of a species q x = ρ x /ρ . The mass fraction of water vapor, q v , is knownas the specific humidity. The mass continuity equationfor each species may be rearranged into a Lagrangianequation for q x , ρ dq x dt = ρ ˙ χ x − ∇ · D x . (30)We may also define the specific internal energy of themixture by u = (cid:88) x q x u x , (31)where u x is the specific internal energy of species x . Fi-nally, we may write the fundamental thermodynamic re-lation for each species, du x dt = T ds x dt − p x dα x dt , (32)where s x is the specific entropy of species x , p x is the par-tial pressure of species x and α x = ρ − x . Combining theprevious two equations, we may write a thermodynamicrelation for the mixture given by (Pauluis, 2011), dudt = T dsdt − p dαdt + (cid:88) x g x dq x dt . (33) Here, the entropy of the mixture s is defined analogouslyto (31), α = ρ − , and the total pressure p is the sumof the partial pressures of each species. The quantity g x = u x + p x α x − T s x is the specific Gibbs free energyof each species, which is closely related to the chemicalpotential.Use of (33) involves some approximation that shouldbe noted. In addition to the assumption that the funda-mental thermodynamic relation is valid for each species,we have assumed that each species has the same tempera-ture T . Furthermore, by expressing the internal energy ofthe mixture as the mass-weighted sum of the internal en-ergy of each of species in (31), we have neglected interfa-cial effects between the different species. Such effects areimportant for understanding the formation of clouds andprecipitation in the atmosphere (Pruppacher and Klett,2010), but they are typically neglected when consider-ing its bulk thermodynamics. Note, however, that wehave made no assumption of chemical equilibrium be-tween species; as we shall see, phase changes outside ofequilibrium are an important irreversible entropy sourcein the atmosphere.As for a single-component fluid, we may derive anequation for the entropy tendency of a multi-componentfluid by substituting the thermodynamic relation (33)into the equation governing the internal energy, which fora multi-component fluid may be written (e.g., de Grootand Mazur, 1984; Gassmann and Herzog, 2015), ρ dudt + ∇· (cid:32) D u + (cid:88) x D x h x (cid:33) = − ρp dαdt + ρ ˙ q rad + ρ(cid:15). (34)This equation is identical to the single-component case(23) but for the appearance of the flux divergence of en-thalpy h x = u x + p x α x owing to molecular motions on theleft-hand side . Combining (33) and (34), and using (24)and (30), we may write a local Eulerian budget equationfor the entropy, ∂ρs∂t + ∇ · ( ρ v s ) + ∇ · (cid:32) D u T + (cid:88) x D x s x (cid:33) − ρ ˙ q rad T = ρ ˙ s fric i + ρ ˙ s heat i + ρ ˙ s diff i + ρ ˙ s chem i . (35)This budget is similar to the entropy budget derivedfor a single component fluid, but it has an additionalterm associated with entropy transport owing to the non-advective flux of mass D x on the left-hand side, andadditional irreversible sources ρ ˙ s diff i and ρ ˙ s chem i associ-ated with non-advective transport of mass and chemi-cal changes, respectively, on the right-hand side (e.g., The heat flux D u is sometimes defined to include these molecularenthalpy fluxes (de Groot and Mazur, 1984). ρ ˙ s diff i = − (cid:88) x D x · (cid:0) T ∇ h x − ∇ s x (cid:1) ρ ˙ s chem i = − (cid:88) x ρg x ˙ χ x T .
The entropy production associated with non-advectivetransport ρ ˙ s diff i may be simplified further using the fun-damental thermodynamic relation written in terms of theenthalpy h x , dh x = T ds x + 1 ρ x dp x , (36)which allows one to write, ρ ˙ s diff i = − D x · ∇ p x ρ x T . (37)Molecular diffusion of a species x induces positive ir-reversible entropy production when it transports thespecies from high partial pressure to low partial pres-sure. This corresponds to the entropy production associ-ated with the molecular mixing of the species within thefluid. B. Thermodynamics of a moist atmosphere
We now discuss more specifically the irreversible en-tropy sources in the atmosphere. To do so, we introducean approximate thermodynamic treatment of a moist at-mosphere following Gassmann and Herzog (2008, 2015)that is similar to other theoretical and modeling studiesin the literature (see e.g., Bannon, 2002; Hauf and H¨oller,1987; Romps, 2008).The atmosphere is taken to be a mixture of water va-por ( v ), liquid water ( l ), solid water ( s ) and “dry” air( d ) containing all the well-mixed gases. In Earth’s atmo-sphere, dry air is by far the most abundant component;typical mass fractions of water vapor q v are no largerthan 3-4%, while typical mass fractions of condensed wa-ter species are orders of magnitude smaller. Dry air andwater vapor are assumed to be ideal gases governed byan equation of state of the form, p x = ρ x R x T, with R x = R ∗ /m x being the gas constant for species x ex-pressed in terms of the molar mass m x and the universalgas constant R ∗ . Condensed water species are assumed tobe incompressible, with negligible specific volume. Thespecific entropy s is then given by s = (cid:80) x q x s x where x = ( d, v, l, s ) and the specific entropy of each constituent is defined (Romps, 2008) s d = c pd ln( T /T ) − R d ln( p d /p ) , (38a) s v = c pv ln( T /T ) − R v ln( p v /p ) + L v T , (38b) s l = c pl ln( T /T ) , (38c) s s = c ps ln( T /T ) − L f T . (38d)Here c px is the isobaric specific heat capacity of each con-stituent, which we take to be constant, and T , p , L v and L f are the temperature, pressure, latent heat ofcondensation and latent heat of freezing, respectively, atthe triple point of water. As noted previously, this for-mulation neglects interfacial effects between species, andthe resultant entropy budget therefore neglects certainirreversible processes associated with the spatial distri-bution of different phases within an air parcel (e.g., theirreversibility of cloud droplets coalescing together). Fur-thermore, by aggregating all non-water constituents ofthe atmosphere into a single species “dry air”, irreversibleentropy production owing to chemical reactions otherthan the phase changes of water is neglected. With thesecaveats in mind, we now consider the entropy sources˙ s chem i and ˙ s diff i for a moist atmosphere.
1. Irreversible phase changes
The irreversibility of phase changes is included withinthe entropy source ˙ s chem i , which we divide into com-ponents associated with evaporation/condensation ˙ s evap i and melting/freezing ˙ s melt i . Sublimation may be consid-ered to be a combination of melting and evaporation.Consider first evaporation and condensation of a liquiddroplet in the air. Denoting the evaporation rate per unitmass by e , we have that e = ˙ χ v = − ˙ χ l , and hence,˙ s evap i = e ( g v − g l ) T .
Positive irreversible entropy production requires g v > g l for net evaporation ( e >
0) and g l > g v for net con-densation ( e < g v = g l , a condition known as saturation. Defining g ∗ lv as the value of the Gibbs free energy of water vapor atsaturation with respect to liquid, we have˙ s evap i = e ( g v − g ∗ lv ) T . (39)Using the definitions of the entropy of water vapor (38b)and the enthalpy of water vapor h v = c pv ( T − T ) + L v ,this may be written (e.g., Pauluis and Held, 2002a),˙ s evap i = − eR v ln( R ) . (40)where R = p v /p ∗ lv is the relative humidity and p ∗ lv ( T ) isthe saturation vapor pressure over a liquid surface and6is only a function of temperature. When the relativehumidity is less than one, the air is subsaturated withrespect to liquid water, and evaporation is irreversible.When the relative humidity is greater than one, the airis supersaturated, and condensation is irreversible. Inthe atmosphere, substantial supersaturation is rare, andcondensation occurs close to phase equilibrium. Evapo-ration, on the other hand, often occurs at relative humidi-ties well below 100%, leading to substantial irreversibleentropy production.A similar derivation may be performed for thesolid/liquid phase transition leading to (Romps, 2008)˙ s melt i = − mR v ln (cid:18) p ∗ lv p ∗ sv (cid:19) , (41)where m is the melting rate per unit mass and p ∗ sv is thesaturation vapor pressure over a solid ice surface. Notethat both p ∗ lv and p ∗ sv are functions of temperature only,and they coincide at the freezing point T . For temper-atures below T , p ∗ lv > p ∗ sv and freezing is irreversible,while for temperatures above T , p ∗ lv < p ∗ sv and meltingis irreversible. In the atmosphere, melting and freezingcan occur tens of kelvins from the freezing point, leadingto a substantial irreversible production of entropy.
2. Diffusion of water vapor
In the atmosphere, the non-advective flux of speciesmass occurs as a result of two processes: 1) diffusivemolecular mixing as a result of the random Brownianmotions of molecules of each species and 2) sedimenta-tion of condensed water particles massive enough to haveappreciable terminal velocities. We consider entropy pro-duction by molecular mixing ˙ s mix i here and return to thesedimentation flux in the following subsection.According to (37), the entropy source due to molecu-lar mixing is proportional to the specific volume of eachconstituent, which for condensed water species may beassumed to be negligible. Of the gaseous species, frac-tional gradients in the partial pressure of water vapor canbe orders of magnitude larger than those of dry air, andthe dominant component of s mix i is that due to the dif-fusion of water vapor down its partial pressure gradient,given by (Pauluis and Held, 2002a; Romps, 2008), ρ ˙ s mix i = R v D v · ∇ ln (cid:18) p v p (cid:19) . (42) By neglecting the specific volumes of liquid and solid water, wehave assumed that the freezing point is independent of pressureand equal to the triple point temperature. This is a very goodapproximation except at very high pressures not experienced inEarth’s atmosphere.
Diffusive mixing of water vapor is particularly strongat the boundaries between clouds and the clear-air en-vironment. Here, the combination of diffusive mixingand evaporation of cloud and precipitation particles pro-duces a transport of water vapor into the environmentthat plays an important role in governing the tropicalrelative humidity (Romps, 2014; Singh et al., 2019).In fact, the entropy source owing to vapor diffusion˙ s mix i has a close connection to that of evaporation ˙ s evap i .Consider a cloud droplet suspended in air with a relativehumidity R . Evaporation from this droplet into the sub-saturated environment results in an irreversible entropysource given by (40). Alternatively, suppose the evapo-ration from the droplet occurs reversibly in a molecularboundary layer surrounding the droplet that is at sat-uration. The water vapor is then diffused away fromthe droplet to the far-field which has relative humidity R . This process results in an irreversible source of en-tropy given by (42). As pointed out by Pauluis and Held(2002a), the entropy production in these two cases is ex-actly the same. Evaporating water and transporting itfrom the droplet to its environment results in the sameirreversible entropy production regardless of the micro-scopic details of the transport process.
3. Irreversible sedimentation of precipitation
Consider an air parcel containing a mass fraction ofprecipitation q p with a sedimentation velocity − v t k rel-ative to air. Here k is a unit vector pointing upwards(antiparallel to the gravitational vector). Generally, itis reasonable to assume that the fall speed v t is equal tothe terminal velocity of precipitation set by a balance be-tween the downward gravitational force and the upwarddrag force on hydrometeors (precipitation particles) ow-ing to friction with the surrounding air.The barycentric velocity of the air-precipitation mix-ture may be written, v = (1 − q p ) v a + q p ( v a − v t k ) , (43)where v a is the velocity of air. Since v a (cid:54) = v , sedimen-tation of precipitation is coupled with a compensatingupward non-advective transport of air with a velocity, v a − v = q p v t k . The above equation implies an upward non-advectiveflux of dry air D d = ρq d q p v t k and water vapor D v = ρq v q p v t k . On substitution into (37), these non-advective fluxes give an irreversible entropy source to-talling (Gassmann and Herzog, 2015), ρ ˙ s sed i = − q p v t T ∂p∂z .
Assuming hydrostatic balance, this may be written, ρ ˙ s sed i ≈ ρg ♁ q p v t T , (44)7where g ♁ is the Earth’s gravitational acceleration. Theentropy source ˙ s sed i is therefore positive when precipita-tion falls to the surface ( v t > d sed = g ♁ q p v t that appears on the right-hand side of (44) represents a dissipation rate associatedwith the loss of gravitational potential energy by fallingprecipitation. Physically, this manifests as frictional dis-sipation due to the upward drag force acting on precipita-tion particles sedimenting relative to the air (Pauluis andHeld, 2002a; Pauluis et al., 2000). In our derivation, how-ever, we do not consider in detail the interface betweenair and condensed water species, and ρ ˙ d sed appears in-stead as the hydrostatic approximation to an irreversiblepressure work (Gassmann and Herzog, 2015).As discussed in section II.B.3, the integrated dissipa-tion ˙ D sed = (cid:82) Ω ρ ˙ d sed dV is a sink in the mechanical en-ergy budget. This may be seen explicitly by consideringthe net rate of work performed by the barycentric flow,which may be written,˙ W K = (cid:90) Ω v · ∇ p dV. (45)Rearranging the barycentric velocity (43) into a re-versible component v rev = v a , associated with fluid mo-tions, and an irreversible component v irr = − q p v t k , as-sociated with the sedimentation of precipitation, we maywrite, ˙ W K = (cid:90) Ω v rev · ∇ p dV (cid:124) (cid:123)(cid:122) (cid:125) ˙ W rev − (cid:90) Ω ρg ♁ q p v t dV (cid:124) (cid:123)(cid:122) (cid:125) ˙ D sed . Here we have used hydrostatic balance to express the sec-ond term on the right-hand side in terms of the gravita-tional acceleration. The first term on the right-hand sidegives the rate of work performed by reversible fluid mo-tions ˙ W rev , and the second term gives the precipitation-induced dissipation rate ˙ D sed . Comparing this equationto (18), we see that, in steady state, ˙ W K = ˙ D fric , and wemay identify the rate of work performed by the barycen-tric flow as being the rate of generation of kinetic energyof fluid motions. The total rate at which reversible workis performed ˙ W rev is larger and represents the work re-quired not just to generate kinetic energy of fluid mo-tions, but also to lift water against the Earth’s gravi-tational field (Pauluis and Held, 2002a; Pauluis et al.,2000).
4. The entropy budget of a moist atmosphere
Combining the results from the previous subsections,we may write an approximate local Eulerian material en- tropy budget for a moist atmosphere, ∂ρs∂t + ∇ · ( ρ v s ) + ∇ · (cid:32) D u T + (cid:88) x D x s x (cid:33) − ρ ˙ q rad T = ˙ s mat i (46)where the material irreversible entropy production isgiven by,˙ s mat i = ˙ s fric i + ˙ s heat i + ˙ s evap i + ˙ s melt i + ˙ s mix i + ˙ s sed i . (47)The terms on the right-hand side represent, from left toright, frictional dissipation of the winds (27), molecularheat diffusion (28), irreversible evaporation and conden-sation (40), irreversible melting and freezing (41), irre-versible molecular mixing (42), and dissipation associ-ated with the sedimentation of precipitation (44).Integrating the above equation and neglecting thetime-tendency term, one obtains the steady-state mate-rial entropy budget for the entire atmosphere (Romps,2008),1 A (cid:90) ∂ Ω A (cid:32) D u T + (cid:88) x D x s x (cid:33) · d S − A (cid:90) Ω A ˙ q rad T ρdV = S mat i , (48)where Ω A represents the volume of the atmosphere, ∂ Ω A represents its boundary (the top of the atmosphere andthe surface) and d S is a surface element oriented withoutward pointing normal. The first integral on the left-hand side represents the flux of entropy out of the at-mosphere associated with the molecular flux of heat andwater species. The right hand side ˙ S mat i gives the to-tal material irreversible entropy production in the atmo-sphere defined ˙ S mat i = 1 A (cid:90) Ω A ρs mat i dV. The total entropy production rate associated with eachprocess given in (47) may be defined analogously.Consider the atmosphere over a saturated liquid sur-face (e.g., the ocean) . Using the relationship s ∗ lv − s l = L v /T , where s ∗ lv is the saturation entropy of water vaporover a liquid surface and L v is the latent heat of vapor-ization, we may write (48) as − A (cid:90) ∂ Ω A (cid:18) F LH + F SH T (cid:19) dS − A (cid:90) Ω A ˙ q rad T ρdV = S mat i (49)where F LH and F SH are the surface latent and sensibleheat fluxes from the surface to the atmosphere, and dS = | d S | . In steady state, conservation of total energy in the For a surface with relative humidity less than one, the surfaceevaporative flux of vapor also involves an irreversible source ofentropy that should be added to ˙ S mat i on the right-hand side. − A (cid:90) Ω A ˙ q rad (cid:18) T − T s (cid:19) ρdV = S mat i , (50)where T s is the surface temperature. In the troposphere,˙ q rad is generally negative, while temperature decreaseswith height, so that the left-hand side of (50) is positiveas required by the second law of thermodynamics.Eq. (50) provides an intuitive perspective on the sec-ond law as applied to the atmosphere. The atmosphereis heated by surface fluxes at the relatively warm sur-face and cooled by radiation in the relatively cold tropo-sphere, thus creating an entropy sink that is balanced byirreversible processes. A key question is then the rela-tive importance of the different processes contributing to˙ S mat i . As highlighted by Pauluis and Held (2002a,b), theirreversible entropy production in the atmosphere is dom-inated by moist processes, including the diffusion of wa-ter vapor ˙ S mix i , irreversible phase change ˙ S cond i & ˙ S melt i ,and dissipation associated with precipitation sedimenta-tion ˙ S sed i . In the next three sections, we will discuss indetail the role played by irreversible moist processes inthe dynamics of convective clouds (section IV), tropicalcyclones (section V) and the general circulation (sectionVI).An important limitation of the derivation leading to(50) is the assumption that all chemical species within theatmosphere have the same temperature. This is a goodapproximation for dry air, water vapor, and clouds, butit is not accurate for hydrometeors with appreciable sed-imentation velocities, which often differ in temperaturefrom their surroundings by several kelvins. While the as-sumption of uniform temperature is made commonly instudies of the atmosphere’s entropy budget, it does notallow for consideration of irreversible entropy productionassociated with heat diffusion between falling hydrome-teors and their surroundings. Bannon (2002) and Goody(2003) derived order-of-magnitude estimates to suggestthat such heat diffusion may contribute significantly tothe entropy budget of moist convection. Bannon (2002)and Raymond (2013) derived equation sets suitable fornumerical modeling that include the relevant irreversibleproduction terms. However, numerical models of the at-mosphere typically do not allow for precipitation to havea different temperature from its surroundings, and to ourknowledge, the magnitude of the entropy production as-sociated with heat diffusion between hydrometeors andtheir surroundings has not been assessed in a comprehen-sive way.
5. The role of latent heating
Latent heat release does not appear explicitly in the at-mospheric entropy budget. This is because phase changesthat occur at equilibrium do not cause a change in en-tropy. Rather, the effects of latent heat in (48) are in-cluded through the entropy input into the atmosphereowing to molecular fluxes of water species. However,many studies of the entropy budget make explicit men-tion of entropy production by latent heat release (e.g.,Peixoto et al., 1991), and this term is found to be a ma-jor contributor to the global atmospheric budget (Pascaleet al., 2011). It is therefore worthwhile to reconcile thisbody of work with the treatment given above.The effects of water on the atmosphere’s entropy bud-get may be taken into account in two complementaryways: an “external” perspective common in early stud-ies of the entropy budget (Peixoto et al., 1991), andan “internal” perspective, first quantitatively applied tothe atmosphere’s entropy budget by Pauluis and Held(2002a). In the external perspective, latent heating istreated as an additional external heat source, similarto radiative heating, but otherwise the atmosphere istreated largely as a single-component fluid as in sectionIII.A.1 (Fraedrich and Lunkeit, 2008; Pascale et al., 2011;Peixoto et al., 1991). In the internal perspective, the at-mosphere is treated as a true multicomponent fluid, withphase changes being cast as mass exchanges between thedifferent components (Pauluis and Held, 2002a,b; Ray-mond, 2013; Singh and O’Gorman, 2016).The connection between these two viewpoints was elu-cidated by Pauluis and Held (2002b), who showed thatthe total irreversible entropy production associated withphase change and water vapor diffusion could be approx-imately written as the sum of two terms, one related tothe latent heating rate, and the other related to the workperformed by the expansion of water vapor in the atmo-sphere. Using this relationship, the entropy budget (48)may be written approximately as, (cid:90) ∂ Ω A D u T · d S − (cid:90) Ω A ˙ q rad + ˙ q lat T ρdV ≈ (cid:90) Ω A (cid:0) ˙ s fric i + ˙ s heat i + ˙ s sed i (cid:1) ρdV + (cid:90) Ω A T dp v dt dV, (51)where ˙ q lat is the net heating rate due to phase changes .This equation is similar to the single-component entropybudget [cf. (26)] with the latent heating rate included asan external heating in addition to that due to radiation,but it includes the entropy production associated withsedimentation of hydrometeors, and a term related to This relationship neglects the temperature dependence of the la-tent heats of vaporization and freezing (Pauluis and Held, 2002b;Romps, 2008).
C. Irreversible entropy production in the ocean
Comparatively few studies have investigated the en-tropy budget of the ocean relative to that of the atmo-sphere, but the formalism developed in section III.A.2is equally applicable to the ocean. The oceanic entropybudget includes irreversible entropy sources owing to fric-tional dissipation ˙ S fric i , heat diffusion ˙ S heat i , diffusion ofmass ˙ S diff i , and phase changes ˙ S chem i . In the ocean,˙ S diff i accounts for irreversible entropy production owingto molecular diffusion of salt from regions of high to lowsalinity, while ˙ S chem i accounts for irreversible entropy pro-duction associated with the melting and freezing of seaice outside of phase equilibrium. Estimates of the entropyproduction by these processes indicate that, while saltdiffusion can be a significant source of entropy in certainregions (Gregg, 1984), both salt diffusion (Shimokawaand Ozawa, 2001) and sea-ice melt (Bannon and Naj-jar, 2018) contribute only a small portion of the totalirreversible entropy production of the ocean as a whole.In contrast to the atmosphere, the ocean’s entropy bud-get may be approximated by that of a single-component fluid, with irreversible entropy production occurring pri-marily through heat diffusion and frictional dissipation.A further difference between the atmosphere and theocean is that the ocean is heated and cooled almost ex-clusively at its upper surface (the exception is the heat-ing owing to the geothermal heat flux, which on Earthis quantitatively small). Heat transport through sensi-ble and latent heat fluxes occurs at the air-sea interface,while the penetration depth of shortwave and longwaveradiation through ocean water is no more than 100 mand a few mm, respectively. The temperature at whichthe ocean is heated or cooled by radiation and turbulentfluxes is therefore very nearly equal to the surface tem-perature. The steady-state material entropy budget ofthe ocean may therefore be written (Bannon and Najjar,2018; Tailleux, 2015),1 A O (cid:90) ∂ Ω O F rad + F LH + F SH T s dS = ˙ S mat i , (52)where F rad is the upward radiant energy flux, F SH and F LH are the sensible and latent heat fluxes from theocean to the atmosphere, and the integral is over theocean surface A O . For simplicity, we have neglected thegeothermal heat flux into the ocean and we have assumedthat the difference between the effective temperature ofprecipitation and runoff and the effective temperature ofevaporation is negligible; Bannon and Najjar (2018) showthat making these approximations has a relatively minoreffect on the ocean’s entropy budget.Building on the work of Tailleux (2015), Bannon andNajjar (2018) derived an observational estimate of theleft-hand side of (52), finding a rate of entropy export − ˙ S e = 1 . − K − averaged over the ocean sur-face area. This estimate is broadly consistent with thematerial entropy production rate of the ocean found inthe modeling study of Pascale et al. (2011), although themodel’s thermodynamic formulation did not include fric-tional heating, and so this was left out of the estimateof irreversible entropy production. Recall that the totalmaterial entropy production of the climate system is ofthe order 35-60 mW m − K − averaged over the Earth’ssurface; irreversible entropy production by the ocean isa very small fraction of this total, implying that the at-mosphere accounts for the bulk of the irreversble entropyproduction in the climate system.Bannon and Najjar (2018) also derived an independentestimate of the material entropy production in the oceanusing estimates of the thermal diffusivity and tempera-ture structure of the global ocean. According to this es-timate, ˙ S heat i ≈ .
86 mW m − K − and ˙ S fric i ≈ .
64 mWm − K − with small contributions from salt diffusion andice melt. Given the difficulty in measuring diffusivitiesin the ocean, the resultant estimate of ˙ S mat i is in ratherremarkable agreement with the estimate of the entropyexport − ˙ S e given above.0The ocean, unlike the atmosphere, is forced boththermodynamically, through surface heat and freshwa-ter fluxes (so-called buoyancy fluxes), and mechanically,through the work done on the ocean by the winds andtides . One application of the oceanic entropy budget isin understanding the relative roles of these two types offorcing on the ocean’s overturning circulation.It is well known that the circulation produced in a fluidheated and cooled at the same geometric level is substan-tially weaker than if the heating occurs below the coolingSandstr¨om (1908) . Some authors have taken this toimply that mechanical forcing plays a dominant role ingoverning the ocean’s overturning circulation (Munk andWunsch, 1998; Wunsch and Ferrari, 2004). But morerecent work notes that buoyancy forcing is an impor-tant source of available potential energy for the globalocean (e.g., Hughes et al., 2009; Tailleux, 2009, 2013, seealso section VIII.A.1), and numerical evidence suggeststhat the ocean’s overturning circulation responds bothto changes in buoyancy fluxes and changes in the atmo-spheric wind field (e.g., Morrison et al., 2011).A key question is how buoyancy fluxes act to alter thekinetic energy generation rate of the ocean. The estimateof the entropy source owing to frictional dissipation ˙ S fric i provided by Bannon and Najjar (2018) is based on theassumption that the frictional dissipation rate is equal tothe work input by the winds and tides. To the extent thatthis budget is closed, it is therefore consistent with thenotion that buoyancy forcing results in no net increasein kinetic energy generation of the oceanic circulation.We note, however, that the estimates of entropy produc-tion given in Bannon and Najjar (2018) are necessarilycrude due to the lack of detailed observations, and theyare not sufficiently accurate to strongly constrain the ki-netic energy dissipation rate. Moreover, as pointed outby Hughes et al. (2009), buoyancy forcing may facilitatea release of kinetic energy at large scales even if does notprovide a net increase in the total kinetic energy gen-eration rate of the ocean. Constraining the magnitudeof such energy transfers between scales using either theentropy or mechanical energy budgets remains an obser-vational challenge. IV. THE ENTROPY BUDGET OF ATMOSPHERICCONVECTION
In the field of meteorology, convection refers to fluidmotions that transport heat in the vertical direction.This is primarily accomplished through clouds and By Newton’s third law, the ocean also performs work on theatmosphere, but this a negligible term in the atmosphere’s me-chanical energy and entropy budgets. See Kuhlbrodt (2008) for an English translation. their associated circulations, from shallow boundary-layer clouds over the subtropical ocean, to explosive con-tinental convection that spans the depth of the tropo-sphere and can potentially produce lightning, hail andother severe weather. Despite its ubiquity in the atmo-sphere, our fundamental understanding of atmosphericconvection and its interaction with planetary-scale flowsremains limited. Basic questions such as what physicalfactors determine cloud updraft velocities are still notcompletely resolved. A key reason for this is the impor-tance of moist processes, including evaporation, latentheating, and precipitation, in atmospheric convection.The complex interaction of moist processes with atmo-spheric fluid dynamics presents a challenging theoreticalproblem.In this section, we describe how analysis of the sec-ond law of thermodynamics has provided a range of in-sights into the dynamics of moist convection. We beginby introducing a common idealized conceptual and mod-eling framework for studying moist convection known asradiative-convective equilibrium (section IV.A), beforereviewing theories of moist convective updraft velocitiesdeveloped based on analysis of the second law (sectionIV.B). Finally, we introduce some new results concern-ing the effect of the “organization” of moist convectionon its mechanical efficiency with the aim of motivatingfurther work in this area (section IV.C).
A. Radiative-convective equilibrium (RCE)
Radiative-convective equilibrium (RCE) describes ahypothetical state in which a surface of infinite extent,usually assumed to consist of liquid water, is held at afixed temperature, and the atmosphere above is allowedto cool under the influence of radiation. The coolingdestabilizes the atmosphere, eventually leading to con-vection. “Equilibrium” (more accurately a statisticallysteady state) is achieved when the convective heat fluxfrom the surface balances the integrated radiative coolingrate (Robe and Emanuel, 1996). RCE is similar to thecanonical fluid mechanics problem of Rayleigh-B´ernardconvection between two plates, but the upper plate isabsent and replaced by bulk cooling of the fluid.RCE provides a starting point for thinking about ver-tical heat transport in an atmosphere without horizon-tal variations. The first studies to calculate RCE solu-tions used it as a model for the tropical-mean or global-mean climate (Manabe and Strickler, 1964; Manabe andWetherald, 1967). While recent studies have constructedanalytical approximations for mean temperature and hu-midity profiles in RCE (Romps, 2014), solutions to thefull turbulent cloud field are only accessible through nu-merical models. But with increased availability of com-putational resources, RCE has become a popular nu-merical and theoretical framework for studying the dy-1namics of moist convection (e.g., Bretherton et al., 2005;Held et al., 1993; Singh and O’Gorman, 2014; Wing andEmanuel, 2014), and in particular, for developing andtesting theories of moist convective updraft velocities(e.g., Pauluis and Held, 2002a; Robe and Emanuel, 1996;Seeley and Romps, 2015; Singh and O’Gorman, 2013,2015). Recent work has also considered the problem ofrotating RCE, in which the atmosphere is assumed to ex-ist on a planet with a finite rotation rate (e.g., Khairout-dinov and Emanuel, 2013; Nolan et al., 2007). In thissection, we will consider only nonrotating RCE, but therotating case will be relevant to the discussion of tropicalcyclones in section V.A disadvantage of RCE is that it does not exist onEarth, and so there are no direct observations with whichto compare numerical solutions. Our discussion in thissection will therefore remain theoretical and modelingbased, but we will discuss estimates of the entropy budgetof Earth’s atmosphere in section VI.
1. Dry radiative-convective equilibrium
While our primary interest is in understanding moistconvection, it is instructive to begin by considering thesimpler case of “dry” convection, in which the phasechange of water plays no role in the dynamics, latentheating is absent, and the enthalpy transport from thesurface is entirely composed of sensible heat fluxes. Wefurther simplify the problem by assuming that the radia-tive cooling rate may be expressed as a function of thetemperature, ˙ q rad = − c pd ( T − T ) τ , (53)where c pd is the specific heat capacity of dry air, T = 200K, and τ = 40 days (Pauluis and Held, 2002a).To demonstrate the RCE state, we ran a simulation ofthe above configuration using a “cloud-resolving” modelof the atmosphere. Cloud-resolving models are numericalmodels that have higher resolutions than typical globalclimate models, allowing them to explicitly represent con-vective clouds on the model grid. However, such modelsremain very far from direct numerical simulation (DNS)of the atmosphere, and they must rely on parameteri-zations for turbulence and other subgrid-scale processes.We discuss the challenges in modeling the second lawusing cloud-resolving models further in section VII.A.More specifically, our RCE simulation was run withthe Bryan Cloud Model (CM1 version 13; Bryan andFritsch, 2002) with the lower boundary fixed at a tem-perature of 301.5 K and without moisture. The domainwas 200 ×
200 km horizontally, with periodic boundaryconditions in both x and y dimensions, and with a hori-zontal grid spacing of 500 m and 64 vertical levels. Themodel was run for 100 days, and statistics were accumu-lated over days 90-100 representing a statistically steady state. Further details of the model configuration are asspecified in Singh and O’Gorman (2013).A schematic of the dry RCE simulation is presentedin Fig. 3a. The radiative cooling (53) produces a tropo-sphere with a depth of roughly 10 km in which vigorousconvection occurs and the domain-mean temperature de-creases with height at a rate close to the dry-adiabaticlapse rate ( ∼
10 K km − ). Above this level, radiativecooling is not sufficient to cause convection, and onlyweak overturning is present.Fig. 4a shows a snapshot of the vertical velocity distri-bution at 4 km; updrafts and downdrafts fill the domain,with magnitudes in the range 2-4 m s − . What sets themagnitude of these updrafts, and how does it depend onthe radiative cooling rate? The entropy budget providesa useful perspective.The entropy budget for the dry RCE case may be writ-ten (section III),˙ Q rad (cid:32) T ˙ Q − T s (cid:33) = ˙ S fric i + ˙ S heat i , where ˙ Q rad = A − (cid:82) Ω A ˙ q rad ρdV is the integrated radia-tive heating rate, expressed in units of W m − by dividingby the area of the domain A , T s is the surface temper-ature, and T ˙ Q is the effective temperature of radiativeheating, defined, ˙ Q rad T ˙ Q = 1 A (cid:90) Ω A ˙ q rad T ρdV. (54)In the atmosphere, we expect frictional dissipation todominate over molecular heat diffusion as an irreversibleentropy source except in a molecular boundary layer im-mediately adjacent to the surface (Pauluis and Held,2002a; Singh and O’Gorman, 2016). The entropy produc-tion owing to molecular heat diffusion may therefore beneglected entirely if we replace the surface temperature T s with the temperature of the air above this molecularboundary layer, T sa (Romps, 2008). The entropy budgetof the dry RCE state then becomes,˙ Q rad (cid:32) T ˙ Q − T sa (cid:33) = ˙ S fric i . (55)Identifying T sa as the input temperature, T ˙ Q as the out-put temperature, and | ˙ Q rad | as the heat input, (55) maybe used to define the Carnot efficiency (section II.B.2)and mechanical efficiency (section II.B.3) of convectionin RCE. This is the rate at which temperature decreases with height asan air parcel is lifted adiabatically and without phase change.
200 220 240 260 280 300 temperature (K) b he i gh t ( k m )
67 dry a he i gh t ( k m ) c
15 106moist
FIG. 3 Schematic of (a) dry and (c) moist radiative-convective equilibrium. Horizontal black lines denote the tropopause ineach case and vertical arrows show the mean sensible (gray) and latent (black) heat fluxes into the atmosphere in W m − . Panel(b) shows mean temperature profiles in the dry (gray) and moist (black) cases, and temperature of a parcel initialized withthe mean temperature and humidity of the lowest model level of the moist simulation and lifted adiabatically while assumingall condensate is immediately removed from the parcel by precipitation (dotted). Mean quantities are calculated as horizontalaverages over the domain and over days 90-100 in each simulation. In a statistically steady state, we expect the rate ofgeneration of kinetic energy by the atmosphere to equalits dissipation. We may write this balance,˙ W K = T fric ˙ S fric i , (56)where T fric is the effective temperature of frictional dis-sipation defined analogously to (54). Here, the left-handside gives the work done by the pressure gradient force inproducing kinetic energy of the winds, and the right-handside gives the frictional dissipation rate.Combining the entropy budget (55) and the mechanicalenergy budget (56) allows one to derive an estimate ofthe vigor of convection (as measured by its rate of work)given the heat input | ˙ Q rad | and estimates of the effectivetemperatures T sa , T ˙ Q , and T fric (Emanuel, 2001),˙ W K = ˙ Q rad T fric (cid:32) T ˙ Q − T sa (cid:33) . (57)The pressure-work ˙ W K may be related, using hydrostaticbalance, to the upward buoyancy flux, which, in turn,may be used to derive a scale for the vertical velocity (seee.g., Emanuel et al., 1994). For dry RCE, the mechanicalefficiency is close to its maximum value, and, all elsebeing equal, the rate of work done by the convective heatengine scales linearly with the heat input | ˙ Q rad | .
2. Moist radiative-convective equilibrium
Let us now consider the case of moist RCE. The dryRCE simulation is rerun, but with the lower boundarycondition assumed to be a saturated surface of water (Fig. 3c). The enthalpy flux from the surface now in-cludes evaporative fluxes of water vapor in addition tosensible heat, and the transport and interaction of thethree phases of water in the atmosphere is accounted for.At equilibrium, the precipitation rate through the lowerboundary is equal to the rate at which water vapor isevaporated into the atmosphere.The simulation highlights a number of important dif-ferences between dry and moist RCE: • In moist RCE, rising air parcels rapidly become sat-urated, leading to condensation and latent heating.As a result, the temperature lapse rate in moistRCE is reduced from dry adiabatic to nearly moistadiabatic (Fig. 3b; Singh and O’Gorman, 2013). • The irreversible fallout of precipitation reduces thewater content of air and allows descending airparcels to be unsaturated. This leads to an asym-metry in upward motion; moist convection favorsnarrow rapid updrafts and broad weak descent(Fig. 4b; Bjerknes, 1938). • Turbulence in moist RCE is weaker than in the drycase (Pauluis and Held, 2002a); in our simulations,the mid-tropospheric vertical velocity distributionis substantially narrower in moist RCE comparedto dry RCE (Fig. 5), despite the fact that the moistcase has a larger heat input | ˙ Q rad | (see Fig. 3). The moist adiabatic lapse rate is the rate at which temperaturedecreases with height for a saturated air parcel lifted adiabati-cally assuming all condensate is immediately removed by precip-itation. FIG. 4 Snapshot of vertical velocity at a height of 4 km and at day 90 in simulations of (a) dry and (b) moist radiative-convective equilibrium. Black contour in (b) shows denotes regions within clouds (cloud water content greater than 0.01 gkg − ). • Finally, Robe and Emanuel (1996) showed thatcloud updraft velocities in moist RCE are virtu-ally independent of the heat input | ˙ Q rad | (see alsoCraig, 1996), in contrast to the case of dry RCE,for which updraft velocities increase with the heatinput (Emanuel et al., 1994). In moist RCE, in-creased radiative cooling is balanced by an in-creased area fraction of clouds rather than anychange in their updraft velocities.These differences suggest that the effects of moisturefundamentally change the dynamics of moist convectioncompared to its dry counterpart. A theory of moist con-vection must therefore account for these differences; it isthe search for such a theory toward which we now turn. B. Theories of moist convection
1. The moist convective heat engine
Emanuel and Bister (1996) attempted to use the en-tropy budget of RCE to derive a theory for moist convec-tive updraft velocities. They focused on the integratedvertical buoyancy flux associated with convection, givenby, F b = 1 A (cid:90) Ω A ρwb dV, where w is the vertical velocity of the fluid, and b = − g ♁ (cid:18) ρ − ρρ (cid:19) is the buoyancy, defined using the horizontal mean den-sity ρ . If the pressure field is in approximate hydro-static balance, the integrated buoyancy flux is approx-imately equal to the rate at which the atmosphere per-forms work in order to generate the kinetic energy ofthe winds F b ≈ ˙ W K (e.g., Romps, 2008). Emanuel andBister (1996) assumed that the buoyancy flux could beapproximated by the buoyancy of an air parcel lifted adi-abatically from the subcloud layer multiplied by the totalcloud mass flux, so that the kinetic energy generation ratemay be written, ˙ W K ≈ F b ≈ M c (CAPE) , where M c is the upward cloud mass flux from the sub-cloud layer, and CAPE is the convective available po-tential energy, defined as the kinetic energy producedby buoyancy forces as an air parcel of unit mass risesadiabatically to the tropopause . Further assumingthat frictional dissipation is the dominant source of irre-versible entropy production in the atmosphere, the aboveequation may be combined with the entropy budget togive a scaling relation for the CAPE,CAPE = ˙ Q rad T fric M c (cid:32) T ˙ Q − T sa (cid:33) . (58)This provides a velocity scaling for cloud updrafts givenby w ∼ √ The temperature of such a parcel is shown in the dotted line onFig. 3. -10 -5 0 5 10 vertical velocity (m s -1 ) -4 -2 P D F ( s m - ) drymoist FIG. 5 Empirical probability distribution functions (PDFs)of the vertical velocity at a height of 4 km in simulationsof dry (gray) and moist (black) RCE. PDFs are constructedfrom hourly snapshots from days 90-100 of each simulation.
Renn´o and Ingersoll (1996) based on a heat engine anal-ysis of an air parcel completing a cycle rising througha cloud and descending through the environment. Sincethe cloud mass flux M c scales with the integrated ra-diative cooling rate ˙ Q rad in RCE, (58) implies that thecloud updraft velocity is only weakly dependent on ˙ Q rad ,as seen in numerical simulations (Robe and Emanuel,1996).A difficulty with both the theories of Emanuel and Bis-ter (1996) and Renn´o and Ingersoll (1996) is that they as-sume that the dominant irreversible entropy productionmechanism in the atmosphere is that of frictional dissipa-tion, thereby neglecting irreversible moist processes. Aswe show below, this is a poor assumption for moist RCE,and it is almost certainly a poor assumption for convec-tion on Earth. A full theory for moist convection basedon the entropy budget requires careful consideration ofthe irreversible entropy production associated with moistprocesses.
2. Moist irreversible processes in radiative-convectiveequilibrium
Pauluis and Held (2002a) were the first to provide acomprehensive estimate of the material entropy budgetof both dry and moist RCE. The authors used simulationsof RCE with a two-dimensional cloud-resolving model inwhich only the liquid-vapor phase transition was con-sidered to provide a detailed evaluation of the relativeimportance of frictional dissipation compared to othersources of entropy. Since then, Romps (2008) and Singhand O’Gorman (2016) have confirmed these results usingthree-dimensional simulations that include the ice phase.Table II shows an example of the entropy budget in a simulation of moist RCE over a liquid water surfaceheld at a temperature of 301.5 K. The simulation dif-fers from the examples shown in Figs. 3-5 in that itallows for the interactive effects of radiation by includingan explicit parameterization of radiative transfer. Thesimulation is identical to the control simulation in Singhand O’Gorman (2016) except that it is run with a hor-izontal grid spacing of 2 km, and on a doubly periodicdomain 288 ×
288 km in size. We include interactive ra-diation and use a larger domain in order to allow the sim-ulation to undergo the phenomenon of “convective self-aggregation”, which is described in the next subsection.The key result of Pauluis and Held (2002a), also ev-ident in table II, is that the entropy source associatedwith frictional dissipation is a relatively small compo-nent of the total irreversible entropy production of RCE,accounting for roughly 15%. This is in stark contrastto dry RCE, in which frictional dissipation is the mainmechanism by which entropy is produced irreversibly.The physical reason for the small value of ˙ S fric i is thatmoist RCE includes a number of additional irreversiblesources of entropy, including entropy production asso-ciated with the sedimentation flux of precipitation ˙ S sed i (Pauluis et al., 2000) and that associated with diffusionof water vapor ˙ S mix i and irreversible phase changes ˙ S evap i and ˙ S melt i (Pauluis and Held, 2002a), which we have com-bined in table II into a single term ˙ S mem i ,˙ S mem i = ˙ S mix i + ˙ S evap i + ˙ S melt i . The sum of all irreversible entropy sources must, insteady state, balance the total radiative sink of entropy − ˙ S mat e . The additional sources of entropy associatedwith phase change, mixing, and precipitation sedimen-tation must therefore occur at the expense of entropyproduction associated with frictional dissipation. In fact,the sources ˙ S mem i and ˙ S sed i are the two largest irreversibleentropy sources in RCE, accounting for the vast majorityof the irreversible entropy production (table II).While similar results to those given above have beenfound using other numerical models (Pauluis and Held,2002a; Romps, 2008), and the dominance of moist irre-versible processes in RCE may be understood on theo-retical grounds (Goody, 2003; Pauluis and Held, 2002a),care must nevertheless be taken in interpreting the en-tropy budget of cloud-resolving simulations. Irreversiblemolecular processes that produce entropy in the atmo-sphere are not resolved in such simulations. Rather, theeffects of these molecular processes must be parameter-ized. Such parameterizations do not always faithfullyrepresent the second law of thermodynamics. For exam-ple, the entropy production associated with heat diffusionis small but negative in our simulation of RCE, appearingto violate the second law. We further discuss the reasonsfor such apparently unphysical entropy sinks, and otherissues relating to accurately modeling the entropy bud-get, in section VII.5 symbol units disaggregated aggregatedentropy budgetexport − ˙ S mat e mW m − K − . ± . . ± . S mat i mW m − K − . ± . . ± . S heat i mW m − K − − . ± . − . ± . S fric i mW m − K − . ± . . ± . S sed i mW m − K − . ± . . ± . S mem i mW m − K − . ± . . ± . − ˙ Q rad W m − . ± . . ± . η M % 1.5 0.5Carnot efficiency η C % 9.2 7.5precipitation efficiency (cid:15) % 25 49TABLE II Entropy budget and other statistics calculated from a simulation of moist radiative-convective equilibrium withinteractive radiation. Simulation is run following the configuration of Singh and O’Gorman (2016), but on a 288 ×
288 kmhorizontal domain and with a horizontal grid spacing of 2 km. Lower boundary condition is an ocean surface held fixed at 301.5K, and simulation is initialized from a state of rest. “Disaggregated” corresponds to mean over day 20-40 of the simulation,when convection remains scattered, and humidity variations across the domain are weak (Fig. 6a). “Aggregated” corresponds toa mean over day 150-230 of the simulation, when the domain consists of a single moist region containing convection surroundedby a dry region devoid of clouds (Fig. 6b). Uncertainties associated with estimating the steady-state budget using a finitetimeseries are quantified using a block-bootstrap method and given as the 90 th percentile confidence interval. In steady state, the frictional dissipation of winds inthe atmosphere must balance their generation by me-chanical work. The above results therefore demonstratethat moist convection has a low mechanical efficiency η M compared to dry convection and compared to the effi-ciency of an ideal Carnot heat engine (table II). Here wedefine the mechanical efficiency similarly to (19), with˙ Q in = | ˙ Q rad | the radiative cooling rate, so that, η M = ˙ W K | ˙ Q rad | . Pauluis and Held (2002a) performed a nondimensionalanalysis of their RCE simulations to show that this me-chanical efficiency depends on the relative importance oflatent heat transport compared to sensible heat transportand the relative importance of the work done by watervapor compared to the total work performed by moistconvection. In moist RCE at temperatures characteristicof Earth’s tropics, latent heat transport is the dominantvertical heat transport mechanism, and water vapor ex-pansion accounts for a substantial fraction of the totalwork. Both of these factors reduce the efficiency of moistconvection.Romps (2008) pointed out that irreversible melting andfreezing further contributes to the low mechanical effi-ciency of moist convection. He also noted that the re-duced lapse rate in moist convection compared to dryconvection (Fig. 3b) results in a smaller temperaturedifference between the input and output temperaturesgiven the same radiative cooling profile. Holding the ra-diative cooling fixed, the moist convective heat engine therefore has a lower Carnot efficiency than its dry con-vective counterpart.An alternative perspective on the reduced mechanicalefficiency of moist convection compared to dry convec-tion is to consider the atmosphere as a combination heatengine and steam cycle. As described by Pauluis (2011),the thermodynamic action of atmospheric convection isto transport heat vertically in the atmosphere, but also todehumidify the atmosphere. These two thermodynamicoperations are in competition with each other, such thatthe dehumidification process reduces the work availablefor the atmospheric heat engine. The magnitude of thereduction in available work associated with this dehu-midification depends on the relative importance of latentheat transport compared to sensible heat transport, andon the relative humidity at which the mixed-cycle engineoperates.The implications of the small mechanical efficiency ofmoist convection are profound. Firstly, it provides anexplanation for the reduced kinetic energy of moist con-vection relative to dry convection as highlighted in thesnapshots in Fig. 4 and the vertical velocity distributionsin Fig. 5. Furthermore, a small mechanical efficiencyis inconsistent with the theories of Emanuel and Bis-ter (1996) and Renn´o and Ingersoll (1996); such theoriesare predicated on the dominance of entropy productionassociated with frictional dissipation within the entropybudget, and they cannot account for the case in whichthe entropy budget is dominated by moist processes. Fi-nally, the presence of moist irreversible sources of entropymeans that the entropy budget no longer places a strong6constraint on the work done by atmospheric convection;a change in the heat input | ˙ Q rad | or its effective tem-perature T ˙ Q may be balanced by changes in the entropysources associated with moisture, rather than those as-sociated with frictional dissipation.
3. The role of mixing and microphysics
Despite the challenges described above, the entropybudget may nevertheless provide guidance toward a the-ory of moist convective intensity. In particular, Pauluisand Held (2002a) argued that the importance of the moistprocesses for the entropy budget implies that moist con-vective updraft velocities may depend strongly on theeffects of condensate on the buoyancy of air and ulti-mately on the microphysical processes that determinecloud and precipitation formation. Additionally, the im-portance of vapor diffusion and irreversible phase changein the entropy budget potentially suggests that the mix-ing of cloudy and non-cloudy air parcels may be of centralimportance to any theory for convective vigor in moistRCE.Indeed, current theories for moist convective updraftvelocities suggest that they are limited by the sedimen-tation velocity of precipitation (Parodi and Emanuel,2009), or that they are determined by the efficiency ofmixing between clouds and their environment (Seeley andRomps, 2015; Singh and O’Gorman, 2013, 2015). In thelatter case, it is argued that the import of subsaturatedair from the environment allows the profile of temper-ature in moist RCE to decrease with height faster thanthat of a moist adiabat, thereby allowing for finite CAPE.At present, estimates of the rate at which this importof subsaturated air occurs must rely on detailed simula-tions with high-resolution models such as those reportedabove. Such simulations do not resolve the molecularmixing processes directly, but rely on turbulence closureswhose validity in the vicinity of clouds is difficult to es-tablish. Whether the entropy budget may be used toconstrain the rate at which cloudy and non-cloudy air ismixed is a potentially important area of future work.
C. Convective organization and the mechanical efficiencyof moist convection
Previous studies of the entropy budget of RCE havebeen limited to cases of disorganized convection, char-acterized by a statistical equilibrium in which convectiveclouds grow and decay quasi-randomly within the domain(Fig. 4; Pauluis and Held, 2002a; Romps, 2008; Singhand O’Gorman, 2016). However, convection in Earth’satmosphere is often organized into larger-scale structuressuch as squall lines, mesoscale convective complexes, ortropical cyclones (Houze Jr and Hobbs, 1982). While the heat engine characteristics of tropical cyclones havebeen the subject of considerable research (see section V),the broader characteristics of the entropy budget of orga-nized moist convection remain relatively unexplored. Inthis section, we provide a preliminary analysis of the en-tropy budget of a particular type of organized convectionto highlight the potential for future research in this area.Under certain conditions, simulations of RCE areknown to spontaneously develop organization in a pro-cess termed “self-aggregation”. In the aggregated state,clouds and convective activity become confined to a smallregion that remains moist, while the rest of the do-main is characterized by a dry troposphere and subsidingair (e.g., Bretherton et al., 2005; Emanuel et al., 2014;Wing and Emanuel, 2014). Studies of convective self-aggregation have shown that it results from feedbacksbetween clouds, water vapor, and radiation that lead toan instability of the disaggregated state to perturbationsin tropospheric humidity (Emanuel et al., 2014). This in-stability is sensitive to the imposed surface temperature(Wing and Emanuel, 2014) as well as other details of themodel formulation, including the resolution and domainsize (Jeevanjee and Romps, 2013; Muller and Held, 2012).Given its sensitivity to model formulation, the idealiza-tion of the framework of RCE, and the long timescaletaken for convection to self-aggregate (see below), theimportance of self-aggregation as a mechanism for con-vective organization on Earth remains debated (Jakobet al., 2019). It is nevertheless of interest to understandhow the thermodynamic characteristics of the aggregatedstate differ from those of the disaggregated state. Forexample, does self-aggregation increase or decrease themechanical efficiency of moist convection?The simulation of moist RCE described in the previoussection initially develops a quasi-steady state in whichclouds occur randomly and relatively uniformly through-out the domain (Fig. 6a). After roughly 50 days, thesimulation begins to aggregate, and after roughly 150days, a new quasi-steady state in which clouds are clus-tered into a small region is obtained (Fig. 6b).We focus our analysis on two quasi-steady periodswithin this simulation in which convection is disaggre-gated (days 20-40) and aggregated (days 150-230), re-spectively. By comparing the entropy budget in theseperiods, table II presents, for the first time, an analysisof how the entropy budget is affected by convective orga-nization. While the initiation of the self-aggregation pro-cess limits the length of the former period, the disagre-gated state is less variable than the aggregated state, andthe uncertainty in the steady-state budget introduced bysuch variability is relatively small in both cases (table II).In our simulation, convective self-aggregation is asso-ciated with a large fractional reduction in the entropyproduction associated with frictional dissipation, imply-ing a reduction in the rate of work performed by moistconvection. Indeed, the mechanical efficiency of moist7
FIG. 6 Snapshots of column relative humidity (colors) and contour of column cloud water content of 0.5 kg m − (black) in asimulation of moist RCE with interactive radiation at (a) day 30 and (b) day 230. Simulation is run following the configurationof Singh and O’Gorman (2016), but on a 288 ×
288 km horizontal domain and with a horizontal grid spacing of 2 km. Lowerboundary condition is an ocean surface held fixed at 301.5 K, and simulation is initialized from a state of rest. convection decreases by a factor of three between the ag-gregated and disaggregated state. The reduction in ˙ S fric i is balanced by an increase in entropy production associ-ated with mixing and phase change ˙ S mem i and a slightdecrease in the total irreversible entropy production ofthe atmosphere. The entropy source owing to the sed-imentation of precipitation ˙ S sed i also decreases with ag-gregation, and this is associated with a doubling of theprecipitation efficiency, defined as the ratio of total con-densation in the atmosphere to the surface precipitation,in the aggregated state compared to the disaggregatedstate.The results above suggest that the RCE atmosphereis a less efficient heat engine when convection is aggre-gated compared to when it is not, but that the efficiencyby which condensation in the atmosphere is converted toprecipitation at the surface increases under aggregation.The reasons for these differences in efficiency are likelyto be related to the large-scale reorganization of the hu-midity distribution that occurs under aggregation. Forexample, the humidity of the near-cloud environment islikely to be higher when convection is aggregated, con-tributing to the higher precipitation efficiency of the ag-gregated state. On the other hand, the variability ofhumidity within the domain also increases under aggre-gation (Wing and Emanuel, 2014), and this may lead toa larger entropy source associated with vapor diffusion,contributing to the lower mechanical efficiency of aggre-gated convection.Further work is required to determine if the above re-sults may be applied to more general forms of organiza- tion in Earth’s atmosphere. We note, however, that thetendency for tropical convection to aggregate has beenhypothesized to increase in a warmer climate (see e.g.,Wing, 2019). Our results suggest that such an increasein aggregation may have implications for the global at-mospheric heat engine under future climate change (seesection VI.B.3 and Lalibert´e et al., 2015). V. THE THERMODYNAMICS OF TROPICALCYCLONES
When planetary rotation is included, simulations ofRCE spontaneously generate one or more rapidly rotat-ing storms analogous to tropical cyclones (TCs) on Earth(Carstens and Wing, 2020; Khairoutdinov and Emanuel,2013; Nolan et al., 2007; Ramsay et al., 2020). TCs arestunning examples of organized deep convection . Theyare characterized by a primary circulation, consisting ofa rapidly rotating vortex around a low-pressure center,and a secondary circulation, consisting of a (mostly) ther-mally direct overturning circulation. The small excep-tion comes from the enigmatic, dry TC eye, in which athermally indirect flow is produced as buoyant dry airis mechanically forced to descend. Outside the dry cen-tral eye, high-entropy moist air ascends in the saturatedeyewall until it reaches the upper troposphere, where it They are variously called hurricanes, typhoons or cyclones, de-pending on the ocean basin in which they occur.
A. Potential intensity theory
It is generally agreed upon that the primary energysource of a TC is the flux of enthalpy from the sea sur-face (Byers, 1944; Emanuel, 1986; Kleinschmidt Jr., 1951;Malkus and Riehl, 1960; Riehl, 1950, 1954) . This flux isdriven by the thermodynamic disequilibrium between thesea surface and the subsaturated air immediately aboveit. Acting like a heat engine, the TC transports energyfrom the warm surface to the cooler troposphere, produc-ing potentially catastrophic winds in the process. Klein-schmidt Jr. (1951) was the first to identify the enthalpydisequilibrium as responsible for the energy flux from thesea, and he provided the first estimate of the maximum For a translation of Kleinschmidt Jr. (1951) see the appendix ofGray (1994). wind speed of a TC. That study and a later attempt byMalkus and Riehl (1960) laid the groundwork for quan-tifying TC thermodynamics, preceding the celebratedmodern-day potential intensity (PI) theory (Bister andEmanuel, 1998; Emanuel, 1986, 1988) for the maximumattainable surface wind speed (or minimum central sur-face pressure) of an inviscid, axisymmetric TC embeddedin a givern thermodynamic environment.Emanuel (1986) explicitly likened a TC to a Carnotheat engine, demonstrating that the same analytic re-sult for potential intensity can be derived from eitherthe equations of motion and the first law of thermody-namics, or from consideration of an idealized thermo-dynamic cycle performed by an air parcel within a TC.The thermodynamic cycle consists of four legs as fol-lows (Fig. 7):
Isothermal expansion: boundary layerair that converges toward the low pressure center ex-pands but stays relatively isothermal, heated by sensibleheat fluxes from the sea surface. Simultaneously, surfacelatent heat fluxes dramatically increase the air’s moistentropy.
Adiabatic expansion: frictional convergence ofboundary layer air near the surface is balanced by ascentin the deep, cloudy eyewall of the TC. This process isroughly slantwise moist neutral: saturated rising parcelsdo not ascend strictly vertically but rather along slop-ing surfaces of constant angular momentum M . Duringsloping ascent away from the TC core, parcels initiallyapproximately conserve their moist entropy s (neglect-ing the mass loss due to precipitation, which is a smallfraction of the air mass), such that surfaces of constant M and s are parallel. Isothermal compression: as air isexhausted farther radially over time, it loses energy radia-tively as it starts to descend, and finally ( adiabatic com-pression ) the parcel slowly subsides back to the boundarylayer along a vortex line. These last two legs do not oc-cur separately in real TCs; rather, air parcels lose bothentropy and temperature as air descends at large radii.However, the artificial separation of the two legs aids inmathematical comparison to a Carnot cycle.According to PI theory, the positive feedback respon-sible for TC intensification is wind-induced surface heatexchange (WISHE, Emanuel, 1986), whereby strongersurface winds induce stronger sea-to-air energy fluxes,which intensify the vortex and lead to stronger surfacewinds. This feedback is eventually arrested and balancedby frictional drag at the surface. TC structure and PItheory were reviewed by Camp and Montgomery (2001);Emanuel (2004, 1991); Wang (2012); recently Emanuel(2018) comprehensively reviewed the full breadth of trop-ical cyclone research.The outer environment restores a parcel’s M and s back to environmental conditions, ultimately throughcontact with the frictional sea surface. With assumptionsabout the storm structure in the free troposphere abovethe boundary layer, chiefly the assumption of slantwisemoist neutrality, one can determine the energetics of the9 a) CLOSED b) OPEN
0% recirculation 𝑄 " , 𝑆 " 𝑄 ’( , 𝑆 ’( , 𝑟 + 𝑀 " , 𝑟 + 𝑄 " , 𝑆 " , 𝑟 + 𝑀 ’( , 𝑟 + 𝑆̇ ’ 𝑄 " , 𝑆 " 𝑀 ’( 𝑆̇ ’ 𝑄 ’( , 𝑆 ’( , 𝑟 + 𝑀 " , 𝑟 + isothermal expansionisothermal compression adiabatic expansionadiabatic compression w a ll o r doub l y - pe r i od i c bounda r y c ond i t i on s FIG. 7 A schematic of two idealized tropical cyclone overturning circulations (radial cross-section). Grey arrows indicate dryair mass fluxes. Black arrows indicate heat Q , entropy S , water content r T , and momentum M fluxes into and out of thedomain. (a) A closed circulation bounded by an outer wall or doubly periodic boundary conditions. (b) A TC in lateral contactwith the rest of the atmosphere. storm needing only to specify the TC’s boundary layerradial structure of s as a function of M and environmen-tal factors like surface air temperature T s and outflow airtemperature T o (Emanuel, 1986). Having specified theseconditions, a theoretical closed loop describing a parcelundergoing cyclic changes in pressure, volume, tempera-ture, and entropy that is closely analogous to a heat en-gine may be defined. This parcel travels the outermostloop of the entire TC circulation, allowing it to experi-ence the largest temperature range and heating/coolingrange, and thus achieve the maximum wind speed v max : v = T s − T o T s C k C D ( h ∗ s − h BL ) . (59)This expression is the outcome of PI theory. Note thatthe first factor on the right-hand side ( T s − T o ) /T s is iden-tical to the Carnot efficiency of a heat engine operatingbetween thermal reservoirs at temperatures T s and T o .The constants C k and C D are bulk surface exchange co-efficients for enthalpy and momentum, respectively. Theatmospheric specific enthalpy h here is typically approx-imated as h = q d h d + q v h v ≈ c pd ( T − T ) + L v q v for dryair heat capacity at constant pressure c pd , latent heatof vaporization L v , mass fraction of water vapor q v , andreference temperature T . The relevant enthalpy dise-quilibrium that drives flux from the ocean to the atmo-sphere is the difference between the saturated enthalpy ofthe sea surface h ∗ s and the enthalpy of the subsaturatedboundary layer h BL .PI theory was further developed by Emanuel (1988,1991) and a complementary approach was provided byHolland (1997). In these theories, heating due to kineticenergy dissipation was not considered, so the thermo- dynamic efficiency in (59) includes the surface tempera-ture T s in the denominator. Bister and Emanuel (1998)showed that frictional dissipation in the boundary layerincreases the lower air temperature, which leads to anincreased potential intensity because the outflow temper-ature T o replaces T s in the denominator of (59). In nu-merical models that don’t include this frictional heatingterm, the original formulation is still appropriate (e.g.,Cronin and Chavas, 2019). Recent work also shows thatthe outflow temperature T o is not a constant (Emanuel,2012; Emanuel and Rotunno, 2011; Fang et al., 2019), sothe constant Carnot efficiency-like term in (59) does notaccurately reflect the range of temperatures at which theatmosphere loses heat.The application of heat engine theory to cyclones iseven simpler for dry fluids. Renn´o et al. (1998) under-took a heat engine analysis for nearly-dry dust devils(for which planetary rotation is unimportant), where thedriving temperature difference was defined as that be-tween the surface and the top of the boundary layer.Mrowiec et al. (2011) showed that the original Emanuel(1986) PI theory, formulated assuming a saturated coreneutral to slantwise convection, was also valid for purelydry TCs that only receive sensible heat from the surface(provided that there is a sufficient WISHE-type feed-back). This works because PI theory incorporates therole of moisture only in the definition of entropy.Potential intensity theory is useful for bounding theupper limit of peak TC wind speeds. However, the ex-pression for v is not equivalent to a calculation ofwork produced by the extremely dissipative TC heat en-gine (Bister et al., 2011). Hakim (2011) demonstrated0that, even for the most extremal closed path consideredby PI theory, the Carnot cycle analogy is only partiallyappropriate; while the isothermal expansion and adia-batic expansion legs were decent approximations of theinflowing and ascending/outflowing air, respectively, thereturn flow was neither isothermal nor adiabatic. Andof course, in a highly dissipative moist TC, (cid:72) T ds doesnot equal the work produced in a cycle, as it would for areversible Carnot cycle.A recent development of PI theory considers a “dif-ferential Carnot cycle” (Rousseau-Rizzi and Emanuel,2019) in which the closed parcel path is taken as aninfinitesimally wide region bounding the extremal over-turning streamline only for the inflowing and ascend-ing/outflowing branches of the circulation. This ap-proach avoids the somewhat unrealistic description ofthe TC descending branch as two separate legs of adi-abatic compression and isothermal cooling, and it betterillustrates that the thermodynamic efficiency of the TC’soverall overturning circulation is unaccounted for in thecalculation of v max . For a calculation of TC work and en-tropy production as a whole, the entire circulation mustconsidered. Finally, a more general, irreversible formula-tion is provided by Renno (2008) that may have broaderapplicability to a range of geophysical vortices. B. Work and entropy budgets of TCs
Consider a steady-state TC, far from equilibrium withits environment. Let the TC system be defined by acylinder centered on the TC center, bounded below bythe ocean surface and above by the top of the atmo-sphere (Fig. 7). Though a developing TC does growin volume of dry air at the expense of the environment(as defined perhaps by something like the size of the ex-panding outflow region), we shall neglect the cyclogene-sis/cyclolysis stages and consider a steady-state matureTC with a fixed amount of dry air in RCE. The TC re-ceives some amount of entropy from the surface, Q in /T in and exports a larger amount of entropy, Q out /T out tospace. Q in is due to sensible and latent heat fluxes fromthe sea surface and Q out is due to radiative cooling. Fora TC in a closed domain (in this context, we mean nolateral fluxes with the rest of the atmosphere, like Fig.7a), Q in = Q out , and the entropy deficit due to loss at thetop of the atmosphere is balanced by irreversible entropyproduction.To evaluate the work performed by TCs in RCE, weapply a procedure of isentropic averaging to the TC over-turning circulation (Mrowiec et al., 2016; Pauluis andMrowiec, 2013; Rossby, 1937). To do so, we introducethe potential temperature θ = T ( p ref /p ) R d /c pd (60)where p ref = 1000 hPa is a reference pressure. The po- tential temperature is conserved for a parcel of dry airthat experiences adiabatic changes in pressure, and itis related to the entropy of dry air by the approximateequation s d ≈ c pd ln( θ/θ ), for some reference potentialtemperature θ . We also introduce the equivalent poten-tial temperature θ e , which is a similar adiabatic invariantfor moist air. The equivalent potential temperature θ e isrelated to the entropy by s = c pd ln( θ e /θ e ) for some ref-erence equivalent potential temperature θ e .Being thermally direct warm-core vortices, TCs arecharacterized by rising high- θ e air and subsiding low- θ e air. Isentropic averaging allows one to recast spatial datainto a spatial dimension and a thermodynamic dimen-sion. The vertical mass flux at each height is binned by adiscretized thermodynamic variable such as θ e . Theseisentropic mass fluxes may then be used to define astreamfunction Ψ( θ e , z ) as a function of equivalent po-tential temperature and altitude that provides a ther-modynamic perspective on the overturning circulation.Pauluis and Mrowiec (2013) first applied this techniqueto statistically-steady disorganized convection in RCE.Mrowiec et al. (2016) subsequently applied the procedureto a TC, where the θ e structure is a rather natural radialcoordinate (Fig. 8a). Both works showed that the bulk ofthe upward mass flux occurred in asymmetric convectiveregions that don’t appear in an Eulerian streamfunction.As a result, the extremal isentropic streamfunction massflux is always higher than the extremal Eulerian stream-function mass flux, by a factor of three or so. This is alsotrue when the TC is axisymmetric, as a comparison of thepeak mass flux in the Eulerian and isentropic streamfunc-tions given in Fig. 8b and c shows. This demonstratesthat the TC upward mass flux does not decrease mono-tonically with radius from the storm center.The isentropic streamfunction may be considered tobe an approximation of the thermodynamic temperature-entropy ( T - s ) diagram; T decreases monotonically withaltitude between the surface boundary layer and thetropopause, and moist entropy increases with increasing θ e , so the isentropic streamfunction is quite similar to a T - s diagram if you flip it upside-down (compare panels 8cand d). Pauluis (2016) introduced a more formal methodto approximate thermodynamic cycles based on Euleriandata by assuming that each closed circuit in the isen-tropic streamfunction is the path of a parcel of air (namedthe Mean Air Flow As Lagrangian Dynamics Approx-imation, MAFALDA). Isentropically averaged variablesof interest may then be interpolated along each closedpath. This is a strong idealization of real parcel motionwithin a TC, which is not expected to exhibit closed par-cel paths in the vicinity of the eyewall. However it allowsfor an approximate calculation of thermodynamic cyclesexperienced by the overturning circulation (Fig. 8d), in-cluding the motions in randomly distributed convectivetowers. The isentropic streamfunction suggests a compli-cated, distinctly not-Carnot-like spectrum of closed paths1 ac bd FIG. 8 Structure and overturning circulation of an axisymmetric tropical cyclone simulation at steady state [data from a 30-daywindow of a TC in a 6,000 km radius domain at 40 ◦ latitude as described in O’Neill and Chavas (2020)]. (a) Equivalent potentialtemperature ( θ e [K], darker shades for higher values) and absolute angular momentum (dashed; contour value increasing withradius). (b) Eulerian overturning mass streamfunction Ψ [kg s − ] integrated radially outward to 800 km; each contour represents10% of the mass flux. (c) Isentropic overturning mass streamfunction Ψ θ e [kg s − ] integrated radially outward to 800 km; eachcontour represents 10% of the mass flux. (d) Temperature-entropy ( T - s ) diagram corresponding to parcel paths in panel (c)using the MAFALDA technique (Pauluis, 2016), where data gaps present in the contouring procedure in panel (c) have beenfilled via linear interpolation. Numbers in panels (b) and (c) give maximum streamfunction magnitude in units of 10 kg s − . in T - s space, which is confirmed using the MAFALDAprocedure [Fig. 8d, and see Pauluis (2016)].The total work produced by an air parcel containing aunit mass of dry air traversing a reversible Carnot cycleis equal to the net heating: (cid:102) W max = − (cid:72) α d dp = (cid:72) T ds .Here s is the entropy expressed per unit mass of dryair, α d = ρ − d is the specific volume of dry air, and the (cid:101) () indicates an integral over a closed parcel path. Wefollow Pauluis (2016) in considering integrals along pathsfollowing the trajectory of dry air as the working fluidrather than the barycentric flow as was done in previoussections. Unlike the definition of ˙ Q in in (16), the netheating in the following development includes dissipativesources, so the Carnot efficiency is an upper bound of themechanical efficiency.In an irreversible MAFALDA loop within a TC, the netheating is partitioned into the ‘Gibbs penalty’ (cid:103) ∆ G andthe reversible work (cid:102) W rev . The Gibbs penalty (Pauluis,2011, 2016; Pauluis and Zhang, 2017) is due to the sys-tematic removal of Gibbs free energy from the TC systembecause water enters the open system via evaporation in subsaturated conditions (low Gibbs free energy) andleaves via precipitation under generally saturated condi-tions (high Gibbs free energy). The reversible work (cid:102) W rev itself is partitioned between the production of kinetic en-ergy (cid:102) W K and the work used to increase the geopotentialof water of any phase (cid:102) W H O , (cid:102) W rev = (cid:102) W K + (cid:102) W H O . Thus (cid:102) W max can be decomposed in the following way (Fanget al., 2019; Pauluis, 2011, 2016; Pauluis and Zhang,2017): (cid:73) T ds (cid:124) (cid:123)(cid:122) (cid:125) (cid:102) W max = − (cid:73) α d dp (cid:124) (cid:123)(cid:122) (cid:125) (cid:102) W K + (cid:102) W H2O − (cid:73) (cid:88) x = v,l,i g x dr x (cid:124) (cid:123)(cid:122) (cid:125) − (cid:103) ∆ G (61)where g x is the Gibbs free energy of water in phase x ,expressed here per unit mass of dry air, and r T is themixing ratio of total water content, equal to the ratio oftotal water mass to dry air mass in the parcel. This is aclosed line integral of the fundamental relation in termsof enthalpy (36).The most efficient parcel path (‘inner core path’) in theTC is the most extremal one that travels from the bottom2of the boundary layer, up the eyewall and to the top ofthe outflow, experiencing the maximum gradient in bothtemperature and entropy. The mechanical efficiency of agiven closed MAFALDA trajectory η M = (cid:102) W K / (cid:101) Q in maybe approximately expressed as (Fang et al., 2019; Pauluisand Zhang, 2017): η M = η C − (cid:102) W H O (cid:101) Q in − (cid:103) ∆ G (cid:101) Q in . (62)Pauluis and Zhang (2017) numerically simulated anidealized three-dimensional TC and calculated a remark-able mechanical efficiency of η M = 0 . η C for the innercore parcel path. Similar values of the mechanic effi-ciency as a fraction of the Carnot efficiency were reportedby Fang et al. (2019) in a more realistic simulation ofHurricane Edouard (2014). They showed that both (cid:101) Q in and (cid:102) W K of the inner core path increased sharply dur-ing a period of strong intensification. The intensificationoccurred as the storm grew and axisymmetrized, while (cid:102) W H O and (cid:103) ∆ G remained relatively stationary. In thenumerical TC experiments of Pauluis and Zhang (2017)and Fang et al. (2019), MAFALDA trajectories that ex-tend to the upper troposphere have a high efficiency, with (cid:102) W K > (cid:102) W H O , (cid:103) ∆ G . These results highlight the potentialutility of calculations of TC work and entropy budgetsfor understanding TC dynamics.The values found for TC mechanical efficiencies givenabove cannot be directly compared with the efficienciesfound in aggregated vs. disaggregated RCE states in sec-tion IV, because the RCE calculations given in table IIare integrated over the domain, while the above TC effi-ciencies are only measured for the parcel path with thehighest efficiency. A comparison of the efficiency of con-vection in rotating and non-rotating RCE has not yetbeen carried out in the literature, but the MAFALDAprocedure could easily be employed in both cases for adirect comparison. C. Open TCs
Though the isentropic averaging procedure producesa closed streamfunction amenable to a work calculation,the TCs studied in the research described above are allopen to the farther environment. Dry air and water sub-stance are exchanged at the system’s lateral boundaries,causing the system to import or export energy and en-tropy laterally. Note that the upper outflow of the TCin Fig. 8b clearly flows beyond the right-hand boundary,and boundary layer air enters from the outer edge, butthe isentropic streamfunction in panel c still appears asa closed circulation. In fact, it has been shown that mostof the condensed water in a TC comes from the lateralconvergence of water vapor, rather than locally from thesea surface (Trenberth et al., 2007). But the MAFALDA procedure can only operate on closed thermodynamic cy-cles. In order to close an isentropic streamfunction undersubstantial lateral exchange with the environment, theaverage vertical velocity is removed from the subdomaincontaining the TC at each vertical level. The integralsare calculated out to a rather limited radius away fromthe TC center (500-800 km) in order for the signatureof the mass flux in the eyewall to remain appreciable inspite of its small volume. An isentropic average that in-tegrates outward to the deformation radius of the stormwould be dominated by weaker convection occurring farfrom the TC eye.TCs have been treated as open systems in thermody-namic studies (e.g., Juraˇci´c and Raymond, 2016; Liu andLiu, 2004; Tang and Emanuel, 2010, 2012), with TC-environmental exchange considered alongside the tra-ditional vertical boundary sinks and sources. Juraˇci´cand Raymond (2016) calculated a moist entropy budgetfor TCs using dropsonde data interpolated to a three-dimensional grid and included a calculation of lateralfluxes of entropy between the tropical cyclone (in a4 ◦ × ◦ storm-following domain) and the environment.Irreversible entropy production was estimated as instan-taneously balancing the entropy sink due to radiativecooling. This assumption could be inaccurate for TCsthat are experiencing a lot of environmental shear (andin general lateral exchange with the environment), whichcan bring in low-entropy air (Tang and Emanuel, 2010)and lead to evaporation underneath the cooling cirruscanopy. A constraint on the environmental exchangewith a highly dissipative TC is that the net entropy outof the TC domain (to space as well as the broader atmo-sphere/ocean system) must be large and positive in orderto keep a TC at steady state. Theoretically, lateral fluxesof entropy out of the TC domain could be negative if ra-diative cooling is sufficiently high; however Juraˇci´c andRaymond (2016) found that such fluxes were positive intheir observed cases.No study has yet attempted a heat-engine analysis ofthe entire lifecycle of a TC, but Tang (2017a,b) constructa simplified framework that illuminates the role of lateralentropy fluxes before and during tropical cyclogenesis.However, their numerical model omits material produc-tion of entropy within the TC domain. The intensifi-cation study of Fang et al. (2019) is another promisingavenue, and it’s possible that heat engine concepts couldbe further brought to bear in the literature seeking tounderstand the annual frequency of TCs globally (e.g.,Hoogewind et al., 2020; Hsieh et al., 2020). Additionally,the relationship between the thermodynamic perspectiveof MAFALDA and the maximum winds and wind field ofa TC has not been explored in detail, and this physicallink may be an interesting one to pursue. In short, thereare many exciting tools and approaches newly availableto probe TCs that exploit the second law of thermody-namics.3 VI. THE GLOBAL CIRCULATION OF THEATMOSPHERE
In this section, we consider the global atmospheric cir-culation from a thermodynamic perspective. We firstdescribe the material entropy budget of the global atmo-sphere and we compare it to the entropy budget of RCE(section VI.A). Next, we consider the global atmosphericheat engine, and we review theories for its meridional en-ergy transport and its response to global climate changeVI.B). Finally, we broaden our perspective to considerthe heat engines of other planets in the Solar Systemand beyond (section VI.C).
A. The material entropy budget of the global atmosphere
Estimating the entropy budget of the global atmo-sphere is challenging; observational studies often em-ploy relatively crude estimates of effective temperatures(Peixoto et al., 1991) that limit the accuracy of the resul-tant estimates of irreversible entropy production. Globalclimate models are able to provide more detailed diag-nostics than those available from observations, but theypresent difficulties of their own. In particular, global cli-mate models are typically run at horizontal grid spacingsof the order of 100 km and they are therefore unable to re-solve convective clouds. Irreversible entropy productionassociated with moist convection, which was describedin detail in the previous section and is known to accountfor a large fraction of the total irreversible entropy pro-duction in the atmosphere (Pascale et al., 2011), mustbe wholly parameterized within a global climate model.The extent to which parameterizations of convection ac-curately represent this entropy production remains un-known. But even assuming that a model’s parameteri-zations accurately reflect the effect of subgrid processeson the model’s resolved grid, the low resolution (Lucariniand Pascale, 2014) and use of simplified thermodynamicformulations (Fraedrich and Lunkeit, 2008; Pascale et al.,2011) within global climate models imply that their en-tropy budgets differ from that of Earth’s atmosphere (seesection VII), and this can make comparison across studiesand models difficult. Finally, while detailed analysis ofindividual models is possible (Pascale et al., 2011), stan-dard outputs from model intercomparison projects onlyallow for the calculation of approximate entropy budgets,and this can lead to difficulty closing the budget (Lemboet al., 2019).Despite the above challenges, there are a number ofbroad features of the atmosphere’s material entropy bud-get that are known with some confidence. The total ma-terial entropy production of the atmosphere makes upthe vast bulk of the material entropy production of theclimate system, which was estimated in section II.A.2to be in the range 35-60 mW m − K − (Lembo et al., 2019), somewhat higher than the estimates of the en-tropy production of RCE given in table II. However, likein RCE, the atmosphere’s entropy budget is dominatedby moist processes. Lembo et al. (2019) found that en-tropy production associated with the hydrological cycle(terms ˙ S mem i and ˙ S sed i in our formulation) accounted for80-90% of the estimated total irreversible entropy pro-duction in an ensemble of state-of-the-art global climatemodels, consistent with theoretical expectations giventhe dominance of latent heat transport in the energy ex-change between the surface and atmosphere at a globallevel (Pauluis and Held, 2002a). Of the component as-sociated with moist processes, the bulk is due to phasechange and vapor diffusion ˙ S mem i ; Lembo et al. (2019) es-timated that the entropy production associated with pre-cipitation sedimentation accounted for only roughly 4-6mW m − K − , consistent with an observational estimateof the dissipation owing to precipitation sedimentationgiven by Pauluis and Dias (2012).Model-based estimates of the entropy production asso-ciated with frictional dissipation of the winds ˙ S fric i varyfrom roughly 6 mW m − K − (Lembo et al., 2019), sim-ilar to values found for disaggregated RCE, up to twicethis value (Pascale et al., 2011), with correspondinglylarge ranges in estimates of the rate of work performedby the atmospheric heat engine and its mechanical effi-ciency. The reason for this wide range is likely due todifficulties in estimating the frictional dissipation rate inglobal climate models (Lembo et al., 2019); such mod-els often have multiple parameterizations that dissipatekinetic energy, and they may or may not include fric-tional heating within their thermodynamic formulation(Pascale et al., 2011). Additionally, kinetic energy thatis both generated and dissipated at scales smaller thanthe model grid is not included in the model’s mechanicalenergy or entropy budgets. Frictional dissipation esti-mated from global climate models should therefore beconsidered to be only the portion of the dissipation thatis associated with the large-scale flow; it is unclear towhat extent one should compare such estimates to thosederived from higher-resolution models such as presentedin section IV.In summary, while quantitative estimates remain un-certain, qualitatively, the entropy budget of the global at-mosphere shares a number of similarities with the simplercase of RCE discussed in section IV. In particular, thedominance of entropy production associated with moistprocesses limits the mechanical efficiency of the globalatmospheric heat engine, and it limits the rate at whichwork is done by the pressure gradient force. As we shallsee below, this fact plays an important role in under-standing the atmospheric heat engine’s response to globalclimate change.4 B. The global atmospheric heat engine
1. A thermodynamic perspective of the global atmosphericcirculation
Despite the long history of research describing the at-mosphere as a heat engine (e.g., Brunt, 1926, sectionII.B), relatively few studies have expressed the atmo-spheric circulation in traditional thermodynamic coor-dinates (e.g., temperature-entropy ( T - s ) space; Lalibert´eet al., 2015). Rather, the global atmospheric circulationis more commonly characterized in terms of the merid-ional mass overturning streamfunction. This streamfunc-tion may be constructed based on an Eulerian average atconstant height or pressure, in which case it quantifiesthe average mass flow in the latitude-height plane. Al-ternatively, an isentropic averaging technique similar tothat described in section V.B but applied to the verti-cal dimension rather than a horizontal dimension maybe used to re-express the streamfunction as a function oflatitude and an entropy-based vertical coordinate (specif-ically, potential temperature). This isentropic stream-function provides a thermodynamic perspective on theglobal atmospheric circulation, and it may be used toquantify the global atmospheric heat engine.An estimate of the traditional Eulerian streamfunctionreveals the three circulation cells known to characterizeeach hemisphere of Earth’s annual-mean circulation: theHadley, Ferrel, and Polar cells (Fig. 9a; the Polar Cell inthe Northern Hemisphere is too weak to be displayed atthe contour interval shown). The Hadley and Polar cellsare thermally direct; their mass fluxes imply warm airrising, cool air sinking and poleward energy transport.The Ferrel Cell, on the other hand, is thermally indirect;its implied energy transport is toward the equator. Sucha cell is able to exist because eddies—motions in the at-mosphere representing departures from a time or zonalmean—produce a poleward energy transport at midlati-tudes that more than compensates for the equatorwardenergy transport implied by the mean circulation. Inspite of the presence of the Ferrel Cell, the total energytransport by eddies plus the mean flow remains polewardat all latitudes.The Ferrel cell does not appear when the streamfunc-tion is calculated using isentropic averaging. The lowerpanels of Fig. 9 show estimates of the streamfunctionbased on mass fluxes averaged at fixed potential tempera-ture (Fig. 9b) and fixed equivalent potential temperature(Fig. 9c) following Pauluis et al. (2008, 2010). As intro-duced in section V.B, the potential temperature may berelated to the entropy of dry air s d via the approximateequation, s d ≈ c pd ln (cid:18) θθ (cid:19) , where θ is a reference potential temperature. Isosur- ° -30 ° ° ° ° a Hadley HadleyFerrel FerrelPolar p r e ss u r e ( h P a ) -60 ° -30 ° ° ° ° b po t en t i a l t e m p . ( K ) -60 ° -30 ° ° ° ° c latitude equ i v . po t en t i a l t e m p . ( K ) FIG. 9 Mean meridional mass overturning streamfunction es-timated from the ERA-Interim reanalysis (Dee et al., 2011)using 6-hourly snapshots for the years 1981-2000 and calcu-lated using (a) pressure, (b) potential temperature, and (c)equivalent potential temperature as a vertical coordinate us-ing the method described in Pauluis et al. (2010). Black con-tours represent clockwise motion and gray contours representanticlockwise motion. Contour interval is 10 × kg s − with zero contour omitted. Numbers give maximum stream-function magnitude in each hemisphere in units of 10 kgs − . Hadley, Ferrel, and Polar cells are labeled in (a). Thickmaroon curves show mean (b) potential temperature and (c)equivalent potential temperature near the surface (at the pres-sure level p = 0 . p s , where p s is the surface pressure). faces of θ are approximately parallel to isosurfaces of dryentropy, and we therefore refer to such surfaces as dryisentropes. The equivalent potential temperature θ e maybe related to the entropy s through a similar equation, s = c pd ln (cid:18) θ e θ e (cid:19) . Since s includes the entropy of both dry air and waterwithin an air parcel, we refer to surfaces of constant θ e ∼ .
2. Theories for the global atmospheric heat engine
A key goal of climate dynamics research is the develop-ment of a theory for the meridional heat transport in theatmosphere. One approach toward achieving this goal isto relate the heat transport F H to the meridional tem-perature gradient through an “eddy diffusivity” K , suchthat, F H ∝ K ∆ T ∆ y , where ∆ T / ∆ y gives a measure of the gradient of tem-perature T in the meridional direction y averaged over a suitable latitude band. The theoretical challenge is tounderstand the dependence of K on the mean thermody-namic state of the atmosphere.Theories for the eddy diffusivity K go back at leasthalf a century (e.g., Green, 1970; Stone, 1972). Of par-ticular note for the present review is the study of Barryet al. (2002), in which a scaling for K was developed bytreating the atmospheric circulation as a heat engine.Barry et al. (2002) assumed that the net atmosphericenergy flux out of the tropics F H could be related tothe frictional dissipation rate associated with large-scaleatmospheric circulations ˙ D LS through an expression ofthe form, ˙ D LS ∝ ∆ TT F H . (63)As in the previous equation, ∆ T represents a character-istic temperature difference across the midlatitude zonewhile T is a characteristic temperature, giving ∆ T /T the form of a heat engine efficiency. The theory is closedusing a mixing length argument, which expresses the dif-fusivity K in terms of the dissipation rate and a charac-teristic length scale over which fluid parcels are displacedby eddies. Barry et al. (2002) showed that, through theappropriate choice of mixing length, their expression forthe diffusivity was able to account for changes in atmo-spheric heat transport in simulations with a comprehen-sive global climate model across a wide range of param-eters.A number of other diffusive theories for the atmo-spheric heat transport have been proposed [e.g., Green(1970); Held and Larichev (1996); Lapeyre and Held(2003); Stone (1972). See also Held (2019) for a recentreview]. Generally, such theories are developed on thebasis of the budget of available potential energy ratherthan entropy (see section VIII.A.1), but recent work byChang (2019) casts both Barry et al. (2002) and Heldand Larichev (1996) in a common entropy-budget focusedframework, showing that they are both limiting cases ofa more general theory for eddy diffusivity.A common feature of many theories for the atmo-spheric eddy diffusivity, including that of Barry et al.(2002), is that they do not explicitly consider the ef-fect of moist processes. Indeed, the relation (63) maybe compared to similar relations used to develop theoriesof atmospheric convection discussed in section IV.A.2 inwhich the rate of work performed by the atmosphericconvection is related to a forcing parameter through athermodynamic efficiency. As we have seen, such theo-ries do not account for irreversible processes associatedwith water in all its phases, but such processes are re-sponsible for the bulk of the material entropy productionin the atmosphere.While there has been some work to adapt theories ofthe atmospheric eddy diffusivity to include moist thermo-dynamics (see e.g., Lapeyre and Held, 2004; O’Gorman,62011), understanding atmospheric heat transport in amoist atmosphere remains an area of active research. Ac-counting for moist processes is particularly important inthe context of global climate change: as the world warms,the concentration of water vapor in the atmosphere is ex-pected to increase by roughly 7% for each kelvin increasein temperature, following the Clausius-Clapeyron equa-tion. This rapid increase in atmospheric humidity clearlymust be taken into account in any theory for the atmo-spheric heat engine in a warming climate.
3. The atmospheric heat engine under climate change
In the last few decades, the climate science communityhas collectively developed a large archive of simulationdata containing projections of global climate change thatis freely available to researchers (Eyring et al., 2016).This archive provides an opportunity to study how theatmospheric heat engine is affected by climate change,at least in the context of global climate models. Whileevaluating the entropy budget is challenging based onlyon the available outputs (Lembo et al., 2019), Lalibert´eet al. (2015) recently developed a technique for accuratelydiagnosing the strength of the atmospheric heat engineusing only standard model outputs.Lalibert´e et al. (2015) consider the fundamental ther-modynamic relation (33), which may be written in termsof enthalpy as,
T dsdt = dhdt − α dpdt − (cid:88) x g x dq x dt . Assuming all phase changes occur at equilibrium , thelast term is only non-zero when water is either added toor removed from the air, and the above relation may bewritten, T dsdt = dhdt − α dpdt + µ dq T dt , where q T is the total mass fraction of water, and µ isa generalized chemical potential. Integrating with massweighting over the atmosphere Ω A and assuming a statis-tically steady state, the time derivative of the enthalpyvanishes, and the above equation may be transformedinto a budget for work done by the atmosphere,˙ W max = ˙ W K + ∆ ˙ G. (64) Evaporation and condensation may be assumed to occur in phaseequilibrium within a microscopic boundary layer of saturationadjacent to cloud droplets/ice particles, with all irreversibilityoccurring via diffusion of water into and out of this boundarylayer (cf. section III.B.2). Melting and freezing, on the otherhand, cannot be treated this way, and irreversibility associatedwith the melt/freeze cycle is neglected in Lalibert´e et al. (2015). where ˙ W K = − (cid:90) Ω A α dpdt ρdV is the rate at which the atmosphere performs work togenerate the kinetic energy of the winds and˙ W max = (cid:90) Ω A T dsdt ρdV, ∆ ˙ G = (cid:90) Ω A µ dq T dt ρdV. Here the term ˙ W max represents the maximum rate ofwork that would be performed by the atmosphere if theonly form of irreversibility were friction (all else beingequal). The term ∆ ˙ G represents a “Gibbs penalty” re-lated to the effects of moisture, and it primarily repre-sents the power required to maintain the hydrologicalcycle (Pauluis, 2011). This budget is similar to (61) insection V.B [it is also analogous to (20) defined in sectionII.B.3], but, because we consider the the fluid velocity tobe the barycentric velocity of the mixture of air and con-densed water rather than the velocity of air, the workdoes not include the work required to lift water (see sec-tion III.B.3). The work required to lift water is includedin the Gibbs penalty term ∆ ˙ G .Lalibert´e et al. (2015) evaluated the terms in (64) ina simulation of global warming conducted with a com-prehensive global climate model. They found that, whilethe maximum rate of work ˙ W max increased under warm-ing, this was offset by an even larger increase in thepower required to maintain the hydrological cycle ∆ ˙ G such that the rate of work done by the atmospheric heatengine in generating winds ˙ W K decreased with warm-ing. A similar reduction in kinetic energy generationand an increased dominance of moist irreversible pro-cesses has been found in a suite of climate projections byLembo et al. (2019), and in idealized simulations of cli-mate warming induced by increased greenhouse gas con-centrations (Lucarini et al., 2010a), increased solar irra-diance (Lucarini et al., 2010b), and increased ocean heattransport (Knietzsch et al., 2015).Lalibert´e et al. (2015) argued that the rapid increasein the power required to maintain the hydrological cy-cle ∆ ˙ G under warming could be related to the rapid in-crease in the moisture content of the atmosphere follow-ing the Clausius-Clapeyron relation. On the other hand,the maximum rate of work ˙ W max is governed by the ra-diative cooling rate of the atmosphere, and this is knownto increase at a more modest rate under global warm-ing (e.g., Allen and Ingram, 2002). The work performedby the atmosphere to generate winds ˙ W K must then de-crease with warming in order to balance (64).The results of Lalibert´e et al. (2015) contrast with arecent estimate of trends in the generation and dissipa-tion of kinetic energy in the atmosphere based on global7
280 290 300 310
SST (K) en t r op y s ou r c e ( m W m - K - ) FIG. 10 Entropy budget as a function of sea-surface tem-perature (SST) in simulations of radiative-convective equi-librium taken from Singh and O’Gorman (2016) and plottedwith log scale. Material entropy export ˙ S mat e (black circles),total irreversible entropy production ˙ S mat i (black line), andirreversible entropy production owing to frictional dissipation˙ S fric i (red triangles), precipitation sedimentation ˙ S sed i (bluesquares) and irreversible phase change & mixing ˙ S mem i (greenpluses). Dotted gray lines show Clausius-Clapeyron scaling,increasing in proportion to the saturation vapor pressure atthe sea surface. Adapted from Singh and O’Gorman (2016). climate models constrained by satellite and in-situ ob-servations, in which it was found that both the kineticenergy and its generation/dissipation rate increased overthe period 1979-2013 (Pan et al., 2017). If accurate, thisresult suggests that recent warming has been associatedwith an intensification of the atmospheric heat engine.It should be noted, however, that the technique of esti-mating the atmospheric state using global climate modelsconstrained by historical observations (known as reanal-ysis) is not well suited to evaluating climate trends (e.g.,Thorne and Vose, 2010), and further work is needed toconfirm the results of Pan et al. (2017) using other meth-ods.The decrease in ˙ W K with warming seen in global cli-mate simulations also contrasts with cloud-resolving sim-ulations of radiative-convective equilibrium, in which therate of work done by the atmospheric heat engine hasbeen found to increase with surface temperature (Romps,2008; Singh and O’Gorman, 2016). This is despite thefact that the moisture content and radiative cooling ratesvary similarly with temperature in both RCE and globalclimate simulations (Jeevanjee and Romps, 2018).Singh and O’Gorman (2016) examined the entropybudget in a series of simulations of RCE over a wide range of surface temperatures. Their results, reproducedin Fig. 10, show that the magnitude of the irreversible en-tropy production terms associated with moist processesroughly scale with the total radiative entropy sink ˙ S mat e ,rather than with the Clausius Clapeyron equation. Thisallows the entropy production associated with frictionaldissipation ˙ S fric i to also increase with warming.The contrasting response of the atmospheric heat en-gine to warming in global climate models compared tohigh-resolution models run in RCE is puzzling and pointsto fundamental gaps in our understanding of the atmo-sphere’s entropy budget. A major difference between thetwo types of studies is one of spatial scale. Global cli-mate models simulate the entire atmosphere, includingthe large-scale circulations that act to transport heatfrom the tropics to the polar regions. But such modelsare generally run with horizontal resolutions too coarseto explicitly represent cloud-scale circulations, and thework ˙ W K includes only the work required to generatecirculations that are resolved by the model grid (see sec-tion VII). Simulations of RCE, on the other hand, aretypically run on domains too small to contain large-scalecirculations, and the work performed is entirely used togenerate moist convection. Future studies applying thetechnique of Lalibert´e et al. (2015) to a wider range ofmodel types, including both realistic and idealized con-figurations, is clearly needed to better understand howcirculations at different scales are affected by climatewarming, and how this is manifest in changes to the at-mospheric heat engine. A further outstanding questionis whether the differences in mechanical efficiency associ-ated with convective organization highlighted in sectionIV.C may contribute to changes in the strength of the at-mospheric heat engine in its response to climate change. C. Heat engines on other planets
The rapidly growing zoo of detected exoplanets, in ad-dition to our quirky companions in the Solar System,continues to astound the imagination. The assumptionsthat lead us to simple models in Earth’s tropospherecan be completely invalid or irrelevant on other plan-ets. Nevertheless, heat engine concepts that considerentropy budgets, appropriately tailored, have been ap-plied to rocky planets in our Solar System (Bannon andLee, 2017; Goody, 2007; Lorenz and Renn´o, 2002; Lorenzet al., 2001; Schubert and Mitchell, 2013; Titov et al.,2007), Jupiter (Lorenz and Renn´o, 2002; Wicht et al.,2019), tidally locked rocky exoplanets (Koll and Abbot,2016) and hot Jupiters (Koll and Komacek, 2018; Readet al., 2016).An important complication in describing the charac-teristics of planetary heat engines is that, unlike for atraditional heat engine operating between two thermalreservoirs, the Carnot and mechanical efficiencies of a8planetary heat engine are not external parameters. Theinput temperature T in and output temperature T out areboth functions of the climate (see section II.B.2). Eventhe effective emission temperature T ∗ e of a planet, definedas the temperature of a blackbody if it were to emit thesame amount of radiant energy as the planet, is a func-tion of its planetary albedo, which is climate dependent.As a planet’s climate changes, both its mechanical effi-ciency and Carnot efficiency may simultaneously change,perhaps substantially. The history of Venus presents apossible example of such behavior; in its early history,Venus has been hypothesized to be water-rich before arunaway greenhouse effect occurred Walker (1975). Ifthere had been an active hydrological cycle, we can hy-pothesize that the mechanical efficiency of the Venusianheat engine (potentially near the Carnot efficiency atpresent) would be much lower in the past to accountfor the attendant irreversible entropy production asso-ciated with moist processes. However, the Carnot effi-ciency itself may have been lower and more Earth-likeif the presence of water clouds and precipitation allowedfor clear-sky patches to cool off the lower atmosphere tospace. Thus mechanical efficiency as a fraction of theCarnot efficiency is a moving target that may obscuredramatic changes in climate.
1. Rocky planets
Schubert and Mitchell (2013) and Bannon and Lee(2017) estimated the Carnot efficiency of the rocky plan-ets with substantial atmospheres (Venus, Earth, Mars,Saturn’s moon Titan). They found that Venus has amuch higher Carnot efficiency than the other bodies con-sidered, at around 24%. Like the better-observed Earth,Mars and Titan are likely to have sedimentation-relatedmaterial entropy production sources that significantly re-duce the mechanical efficiency of the climate.Bannon and Lee (2017) used a creative approach toestablish an upper bound on the Carnot efficiency and tostudy the importance of spatial variations of absorptionand emission temperatures for the planetary heat engine.Using a variant of the transfer climate system definition(see section II.A), they defined the entropic absorptiontemperature T abs by,˙ Q abs T abs = 1 A (cid:90) Ω ˙ q abs T ρdV, where ˙ q abs is the heating of the climate system by short-wave radiation, with˙ Q abs = 1 A (cid:90) Ω ˙ q abs ρdV. (65)The entropic emission temperature T e was defined anal-ogously as a function of ˙ Q e , based on the temperature at which longwave radiation is emitted directly to space.Unlike (16), (65) does not include net heating by long-wave radiation due to radiative exchanges within the cli-mate system; it is purely defined based on emission ofphotons that leave the Earth-system. Bannon and Lee(2017) showed that T e must be equal to or greater thanthe corresponding “effective” emission temperature T ∗ e necessary for a corresponding blackbody planet of equiv-alent albedo to be in radiative equilibrium with its star.Given a fixed stellar flux, orbital radius and planetarysize, T ∗ e is only a function of the planetary albedo. Em-ploying a thought experiment of the entropy balance oftwo idealized planets, Bannon and Lee (2017) demon-strated that the maximum entropy production of a planetoccurs when T e = T ∗ e , which can be understood as thelowest temperature possible for the heat sink to space.When T e instead varies across the planet, entropy pro-duction decreases. In contrast, a more uniform absorp-tion temperature T abs , or more uniform albedo, decreasesentropy production. For a fixed T abs , maximum entropyproduction occurs when T e = (3 / T abs . Bannon and Lee(2017) further scale these temperatures by a truly exter-nal temperature T bb , which is the temperature the planetwould have if it were a blackbody with zero albedo. Thecorresponding material entropy production can be scaledsimilarly. This permits a nondimensional comparison ofentropy production between planets in a two-dimensionalphase space of T e /T bb and T abs /T bb , and it neatly demon-strates that Venus is indeed unique for its high Carnotefficiency, which is close to the upper bound.Venus has a high albedo (around 76%) and thick sul-furic acid cloud cover, such that little shortwave radi-ation reaches the surface. Venus’ thin thermally directoverturning circulation likely does not penetrate to thestable atmospheric layer closest to the surface—it largelyoccurs aloft in the cloud region, 50-55 km, away from asolid frictional surface. Evidence suggests a possibilityof convective overshooting in Venus’ atmosphere, whichwould indicate a Venusian counterpart to Earth’s hydro-logic cycle (e.g., Baker et al., 1998; McGouldrick andToon, 2008) and would suggest that the mechanical effi-ciency of the Venusian atmosphere is substantially lowerthan the Carnot efficiency. But dissipation due to fric-tion around hydrometeors was estimated to be unimpor-tant by Lorenz and Renn´o (2002). The dominant dis-sipation mechanism that removes mechanical energy ishypothesized to be breaking internal gravity waves (Iza-kov, 2010). Taken together, the studies of Bannon andLee (2017); Lorenz and Renn´o (2002); Schubert et al.(1999) suggest that Venus has a very high absolute me-chanical efficiency, but the role of moist convection is toopoorly understood to have high confidence.Mars, uniquely in the Solar System, has sporadicglobal dust storms that nearly shield the surface fromthe sun. This dust absorbs both shortwave and long-wave radiation, adding to the longwave absorption from9the CO atmosphere. As the dust sediments back to thesurface, it is also likely to be an important source of dis-sipation analogous to the dissipation of falling hydrome-teors discussed in section III.B.3. Given the small differ-ence between Mars’ estimated average absorption tem-perature and emission temperature in the vertical, morework may potentially be produced during horizontal en-ergy transport than vertical transport, which the exis-tence of global dust storms and small dust devils seemsto support (Jackson et al., 2020; Schubert and Mitchell,2013). The dominance of Earth’s vertical production ofirreversible entropy cannot be assumed of other plan-ets. In particular, planets with shallow atmospheres andsmall surface-emission layer temperature differences maystill have a strong lateral temperature gradient. This canlead to increased entropy loss to space, coincident withincreased work production and frictional dissipation inthe atmosphere. Koll and Abbot (2016) show the tem-perature difference relevant to the heat engine model ofa tidally locked rocky exoplanet is the permanent hori-zontal day-night temperature gradient and not the localvertical lapse rate.The most Earth-like planet in terms of entropy sourcesand sinks is likely Saturn’s moon Titan. It has an activemethane cycle with some resemblance to Earth’s hydro-logical cycle, which is a source of irreversible entropy gen-eration due to drag around ‘rain’-drops. Similar to Marshowever, there is a small difference between the aver-age absorption and emission temperatures, rendering arather low estimated Carnot efficiency of 4.1% (Schubertand Mitchell, 2013). It is possible that the general cir-culation is able to produce more work than this Carnotefficiency would suggest if Titan has substantial seasonalcross-hemispheric heat transport, which was argued us-ing energy balance and numerical modeling by Mitchell(2012).
2. Giant planets
Rocky planets can be generally assumed to be in en-ergy balance with their star. They are much smaller onaverage than the fluid planets and are either geologi-cally ‘dead’ with a cold core, or the climate system isshielded from an active core by an insulating rocky man-tle, as in the case of Earth. Either way, the geothermalheat flux on the rocky planets is typically an insignifi-cant fraction of the energy received from the sun. Theirwell-defined solid surfaces bound the climate system ontimescales shorter than the evolution of the lithosphere.These traits make single- and multi-layer energy balancemodels tractable and useful. Giant planets are a muchtrickier system. Gas and ice giants, like Jupiter, Saturn,Uranus and Neptune, are likely to lack any kind of solidlower boundary that could provide a large frictional dragon winds, and modeling efforts do not always carefully consider the need for a physically motivated dissipationmechanism (Goodman, 2009). In the absence of solidfrictional surfaces, leading dissipation mechanisms to bal-ance mechanical energy generation by thermally directflows include turbulence and fluid instabilities, shocks(Dobbs-Dixon and Lin, 2008; Li and Goodman, 2010),Ohmic dissipation (Batygin and Stevenson, 2010) andmagnetic drag (Perna et al., 2010).The idealization of the atmospheric circulation as athermally direct circulation similar to a Carnot engineis a reasonably good model for atmospheric layers withapproximately adiabatic lapse rates, such that convectivemotions can move air quasi-adiabatically in the verticalas they advect heat downgradient. This is certainly agood model for the Earth’s troposphere, but the Earth’sstratosphere—and atmospheres in approximate radia-tive equilibrium generally—do not exhibit thermally di-rect quasi-adiabatic motions (Koll and Komacek, 2018).The Brewer-Dobson overturning circulation in the sta-bly stratified stratosphere, for example, is mechanicallyforced (Haynes, 2005), and it is thought that regionswithin the giant planets, as well as brown dwarf plan-ets, should exhibit thermally indirect, wave-driven over-turning circulations (Showman and Kaspi, 2013), wherethe waves may be excited by convective (thermally directoverturning) activity in an adjacent atmospheric layer.Even in regions of the atmosphere that are character-ized by thermally direct circulations, the Carnot cycle isnot necessarily a good model, particularly for heavily ir-radiated giant exoplanets. These so-called ‘hot Jupiters’,believed to be in abundance throughout the universe,have extremely short and rapid orbits around their star.The thermally direct overturning circulation occurs ap-proximately within an isothermal layer column-wise, witha strong day-night gradient (where the day side is per-manently irradiated because it is tidally locked to alwaysface the star). Koll and Komacek (2018) modeled theheat engine of hot Jupiters instead as an Ericsson cycle.Like the ideal Carnot cycle, an Ericsson cycle is heatedand cooled during isothermal processes, but the othertwo legs are isobaric instead of isentropic. An ideal Eric-sson cycle has the same efficiency η C , but the mechanicalefficiency of hot Jupiters could be larger or smaller de-pending on the unknown potential role of precipitationduring the cycle (see section II.B.3). Hot Jupiters arelikely to host multiple layers of cloud decks (includingsilicate and titanium dioxide clouds) as well as hydrocar-bon hazes (Gao et al., 2020).Lastly, planets need not even be close to energy bal-ance. This is observed for Jupiter, Saturn and Neptune,which emit around 80%-160% more energy than they re-ceive from the sun (Conrath et al., 1989; Ingersoll, 1990;Li et al., 2018). These planets are still cooling, shrinkingand stratifying from their violent formation (Hubbard,1968). Given Q out > Q in , and assuming that average T in > T out as on Earth, we could assume that these plan-0ets are in entropy balance and determine that they mustbe producing additional entropy irreversibly to balancethe enhanced entropy export Q out /T out to space. Butwhy should we assume entropy balance under these con-ditions? More likely, these planets are secularly ‘order-ing’ as well as cooling, and accordingly losing net entropyto space. If the planets are indeed out of entropy balance,how could we measure that remotely? On a fluid planet,can we distinguish between a plausibly fast atmosphericadjustment to entropy balance and a core region thatloses entropy over time? The fluid behaviors (includingmagnetohydrodynamics at depth) of the giant planets arestill extremely unconstrained, so this is an area ripe forfurther observational missions.Lucarini (2009) and Bannon and Lee (2017) call forfuture climate models to routinely calculate and provideemission and absorption temperatures, in order to makepossible quantitative heat engine analyses between simu-lations. As GCM capabilities improve in realism as wellas flexibility (i.e., GCMs specially built for Mars, Jupiter,hot Jupiters, etc.), there is now potential for compara-tive planetary climatology to tackle the evolving Carnotand mechanical efficiencies of ancient and alien worlds.This approach may help bound the possible range in cli-mates of exoplanets, which for the foreseeable future canbe studied only as point sources of light. VII. MODELING THE SECOND LAW OFTHERMODYNAMICS
Given the impracticality of conducting controlled ex-periments on the climate system and the sparseness ofour networks for observing the atmosphere, ocean, andland surface, numerical models represent an essentialcomponent of the climate researcher’s toolkit. Climatemodels are used as numerical laboratories to test hy-potheses about how the climate system operates, as stateestimation tools to study aspects of the climate systemthat go beyond those accessible to observations, and astools for projecting future climate change. Such mod-els implement numerical approximations to physical lawsincluding conservation of energy, mass, and momentumin order to solve for the evolution of the system. Asdiscussed in section III, these conservation laws, com-bined with a suitable definition of entropy, are sufficientto specify the entropy budget. But developing models ofthe climate that produce a realistic entropy budget thatsatisfies the second law of thermodynamics remains chal-lenging. In this section, we discuss some of the issuesraised when attempting to model the second law in thecontext of two types of climate models: cloud-resolvingmodels (section VII.A) and global climate models (sec-tion VII.B).
A. Cloud-resolving models
Cloud-resolving models are numerical models of theatmosphere with horizontal grid spacing (cid:46) . For example, section IV presents theresults of idealized simulations of RCE using a cloud-resolving model on a domain roughly 200 ×
200 km insize.Although their horizontal grid spacing is much smallerthan grid spacings typical for global climate models,cloud-resolving models remain far too coarse to be usedfor direct numerical simulation (DNS) of the atmo-sphere . As a result, cloud-resolving models do notexplicitly resolve the diffusive molecular fluxes of heat,water, and momentum that are involved in irreversibleentropy production in the atmosphere. Rather, the ef-fects of these molecular fluxes must be approximated bythe model’s subgrid-scale turbulence parameterizations.Such parameterizations assume that the turbulent cas-cade of kinetic energy toward the molecular scale maybe expressed in terms of the model’s resolved-scale flow.If this assumption is satisfied, the frictional dissipationrate, and the associated irreversible entropy production,implied by the parameterized subgrid-scale momentumtransports provide a good approximation to the dissipa-tion rate and irreversible entropy source owing to vis-cosity in the atmosphere (Romps, 2008). In other words,we expect subgrid-scale momentum transport to produceentropy irreversibly, thereby satisfying the second law ofthermodynamics (Gassmann and Blender, 2019).In our simulation of RCE described in section IV,subgrid-scale fluxes of momentum are indeed associ-ated with positive frictional dissipation and positive irre-versible entropy production (see table II). Subgrid-scalefluxes of heat, however, are associated with a small butsystematic sink of entropy. As has been noted by previ-ous authors (Gassmann and Herzog, 2015; Goody, 2000;Romps, 2008), parameterized turbulent heat transportdoes not necessarily produce an entropy source of the Recent advances in computing technology are beginning to allowfor global-scale weather and climate models that approach cloud-resolving resolutions (see e.g., Stevens et al., 2019). DNS has been applied to understand the detailed dynamics ofcloud entrainment (e.g., Mellado et al., 2018) and microphysics(e.g., Vaillancourt et al., 2002), but computational constraintscurrently limit the accessible Reynolds numbers far below therequirements for even a single cloud life-cycle, let alone the globalatmosphere. .In numerical models, parameterized turbulent heattransport does not occur in isolation; rather, it is associ-ated with turbulent transports of momentum and mass.Previous authors have argued that the negative entropyproduction owing to parameterized turbulent heat trans-port may be reconciled with the second law by recogniz-ing that turbulent heat transport and turbulent dissipa-tion of kinetic energy are different aspects of the sameturbulent cascade (Akmaev, 2008; Priestley and Swin-bank, 1947). According to this view, the second lawis satisfied provided that the total entropy productionassociated with all turbulent transports is positive. Inshear-driven turbulence layers, it may be shown thatthis condition is guaranteed if the Richardson number(a nondimensional ratio representing the relative impor-tance of buoyancy compared to shear) is below a criticalvalue. In many turbulence parameterizations, the criti-cal Richardson number is taken to represent the onset ofshear-driven turbulence (e.g., Lilly, 1962), ensuring thatsubgrid-scale turbulent heat and momentum transportsonly occur when they would result in a net positive ir-reversible entropy production as required by the secondlaw.Gassmann and Herzog (2015) and Gassmann (2018)have recently argued against the above view, suggestinginstead that the second law requires positive entropy pro-duction for both parameterized turbulent heat transportand parameterized turbulent momentum transport indi-vidually. The authors develop a formulation of turbulentheat transport that satisfies this constraint by includinga subgrid work term in the mechanical energy equation. Traditional parameterizations of subgrid-scale water vapor trans-port can also lead to local entropy sinks because of the work re-quired to diffuse water vapor vertically (Gassmann and Herzog,2015).
Effectively, this formulation shifts the energy source ofwork done by turbulence against the stratification fromthe internal energy of the fluid to the resolved-scale mo-tion. Gassmann (2018) provides evidence that this for-mulation allows for a more realistic simulation of a drygravity wave, but it is at present not widely adoptedwithin the field. Clearly, continued research is needed tofurther clarify the requirements placed by the second lawof thermodynamics on the formulation of turbulence pa-rameterizations used in cloud-resolving models (see e.g.,Gassmann and Blender, 2019).
B. Global climate models
Global climate models, also known as general circu-lation models, are numerical models of the atmosphere,land, and ocean, that are used for both weather predic-tion and climate projection. Because they must cover theentire planet, global climate models are typically run atlower resolution than cloud-resolving models discussedabove, and their horizontal grid spacing within the at-mosphere ( (cid:38)
20 km) is too coarse to resolve convectiveclouds. Since clouds and their associated circulations areresponsible for a large portion of the irreversible entropyproduction in the atmosphere, evaluating and interpret-ing the entropy budget of a global climate model presentsa particular challenge.Johnson (1997) was one of the first to explicitly link theentropy budget of global climate models to biases in theirsimulation of the atmosphere. The author argued thatnumerical dissipation in climate models leads to an artifi-cial source of entropy that spuriously increases the mate-rial entropy production of the simulated atmosphere. Inorder to maintain a steady state, an opposing error in thesimulated entropy import must also be present. Johnson(1997) suggested that this could occur via a cold bias inthe model’s temperature field, providing an explanationfor “the general coldness of climate models”. As pointedout by Lucarini and Ragone (2011), however, a bias inthe entropy import to the atmosphere is just as likely tobe associated with a bias in the radiation field as in thetemperature field. Moreover, the cold bias referred to byJohnson (1997) is much reduced in more recent genera-tions of global climate models (Flato et al., 2013).Woollings and Thuburn (2006) investigated numericalentropy generation in climate-model simulations of a dryatmosphere in the absence of radiative heat transport orheat exchange with the surface—effectively a thermody-namically isolated atmosphere. The authors found bothpositive and negative numerical entropy sources, con-tradicting Johnson’s assumption that numerical entropygeneration acts solely to increase the total internal en-tropy production of a simulated atmosphere. Moreover,numerical entropy sinks within a fluid that is otherwiseisolated correspond to a local violation of the second law2of thermodynamics. To prevent such occurrences, Liuand Liu (2005) suggested an ad-hoc procedure in whichthe diabatic heating rate within a model is altered toensure consistency with the second law.A more rigorous solution is to employ numericalformulations of the governing equations that main-tain their Poission-bracket structure upon discretiza-tion (Gassmann and Herzog, 2008), thereby reproduc-ing exact conservation of energy and entropy (in the ab-sence of non-conservative terms), and avoiding the prob-lem of artificial numerical sources of entropy (Gassmannand Blender, 2019). But while such numerical formula-tions are beginning to be used in cloud-resolving models(Gassmann, 2013), they are not typically used in globalclimate models, which still suffer from numerical errors intheir energy and entropy budgets (Lucarini and Ragone,2011). Many global climate models also employ simpli-fied thermodynamic formulations that neglect processessuch as the heating owing to frictional dissipation (Pas-cale et al., 2011). Care must be taken to evaluate theentropy budget of such models so that it may be com-pared meaningfully to that of the Earth (Pauluis andHeld, 2002a).As in the case of cloud-resolving models, irreversibleentropy production in global climate models is not mod-eled explicitly; it occurs within parameterizations thatcalculate the effect of processes that occur at subgridscales. Because of their low resolution, global climatemodels require parameterizations for processes such asocean mesoscale eddies and atmospheric convection thatare not required by higher-resolution models. Developingaccurate parameterizations for these processes remainsan ongoing challenge. For example, errors in moist con-vection parameterizations have been argued to be respon-sible for long-standing biases in the tropical precipitationdistribution simulated by global climate models (e.g.,Oueslati and Bellon, 2013). Evaluating the ability of pa-rameterizations of moist convection to accurately repre-sent the second law of thermodynamics therefore providesa potential pathway toward their improvement. However,because convection parameterizations represent both re-versible processes (e.g., the generation of kinetic energyby cloud motions), and irreversible processes (e.g., vapormixing and irreversible evaporation at the cloud edge)within the atmosphere, evaluating their compliance withthe second law remains a nontrivial theoretical challenge(Gassmann and Herzog, 2015).The limitations in the representation of the second lawin global climate models, and in cloud-resolving modelsas discussed in the previous subsection, lead to errors intheir simulation of the climate. Such errors are likely tobe quantitatively small, but if they are systematic, theymay nevertheless be consequential for the mean climateand its statistics. For instance, Singh and O’Gorman(2016) reported that the irreversible entropy source dueto vapor diffusion in simulations of RCE was strongly sensitive to vertical resolution, and this resulted in reso-lution dependence of the simulated mechanical efficiencyof moist convection. Given these potential sensitivities,it is our view that a firm theoretical foundation for therepresentation of the second law of thermodynamics inclimate models should be a goal of model developers.Increased availability and use of thermodynamic diag-nostics for the evaluation of climate models, as recentlyadvocated by Lalibert´e et al. (2015) and Lembo et al.(2019) provides one possible step toward this goal.
VIII. VARIATIONAL APPROACHES FOR CLIMATE ANDGEOPHYSICAL FLOWS
The second law of thermodynamics implies that an iso-lated system evolves toward a state of maximum entropy.The eventual state of such a system may therefore besolved through an extremization procedure using varia-tional methods. While Earth’s climate is not isolatedand exists far from equilibrium, similar variational ap-proaches have nonetheless found a range of applicationsin the literature.In particular, variational methods have been used todefine measures of the amount of energy “available” todo work on the climate system. This literature involvesan extremization of a particular energy reservoir underthe constraint that the total energy of the (presumedisolated) climate system is constant. We discuss two ex-amples of such measures in section VIII.A.Furthermore, certain long-lived coherent structures inplanetary fluids exist in an “inertial” regime in whichboth forcing and dissipation are weak. Such flowsare amenable to analysis through statistical mechanicstechniques, despite the forced-dissipative nature of thebroader climate system. We discuss statistical mechan-ical approaches to geophysical fluid dynamics (GFD) insection VIII.B.Lastly, a controversial hypothesis due to Paltridge(1975) extends the idea of entropy maximization toforced-dissipative systems by arguing that such systemstend to maximize their entropy production rate: this isthe Maximum Entropy Production (MEP) principle. Wecritically examine the MEP principle in section VIII.C,concluding that its physical basis remains unclear, and itsapplication to the climate system remains speculative.
A. Entropic energies
The first law does not distinguish between heat andwork, but the second law breaks that symmetry; accord-ing to the second law, work can be completely convertedto heat but heat cannot be completely converted intowork. Thus the second law of thermodynamics indicatesthat the universe is irreversibly and monotonically trans-3forming energy from other forms into unusable internalenergy. In previous sections, this principle was expressedin terms of entropy, but it may also be expressed in ener-getic terms by defining a measure of the energy available to drive motions in a fluid. In this section, we exploretwo parallel threads of research that seek to provide adefinition of such an energy measure.Section VIII.A.1 discusses a common approach in at-mospheric science and physical oceanography, that ofquantifying the Available Potential Energy (APE) of theclimate system as a source of kinetic energy. SectionVIII.A.2 then describes exergy as an alternative and moreformal measure of departure from thermodynamic equi-librium and briefly reviews some applications to the cli-mate system. The reader is referred to Tailleux (2013) fora detailed review of APE, exergy and related concepts.
1. Available potential energy
Margules (1905) and Lorenz (1955) pioneered thequantification of an atmosphere’s ability to drive motion(McWilliams, 2019). Lorenz defined the Available Poten-tial Energy (APE) of the atmosphere A as the componentof the total potential energy P that may be converted tokinetic energy of the general circulation. Evaluating theAPE for Earth’s atmosphere, Lorenz (1955) found it tobe a very small fraction of the total potential energy P .The concept of APE is easy to illustrate. Imagine awater glass containing hot water above cold water with atilted interface between the two water masses (Fig. 11a).At an initial time t , the water is at rest and in hydro-static balance. If the system is allowed to spontaneouslyevolve, one may intuit that there will be rapid turbulentmotion, as the water masses reduce the interface slope tozero. Thus, even though each column individually wasin hydrostatic balance initially, the horizontal gradientof density provides a reservoir of potential energy thatcan be spontaneously converted to kinetic energy: it is‘available’. After sufficient time has passed the turbulentmotions cease and both the hot water and cold watermasses are slightly warmer from frictional dissipation ofthe kinetic energy. But the potential energy is lower thanthat of the initial state . Had the glass been stably strat-ified from the beginning, without horizontal gradients, itwould have remained motionless, maintaining its initialpotential energy. APE provides us with a mathematicaltool to determine when and how much potential energymay be released by spontaneous fluid motion. As translated by Abbe (1910). In the Boussinesq limit, there is no internal energy reservoir,and instead frictional dissipation of motion returns energy tothe potential energy reservoir. Thus if the fluid considered wereBoussinesq, the potential energy of the initial state and finalstate is the same.
HOTCOLD HOT’COLD’ b) 𝑃𝐸(𝑡 % ) = 𝑃𝐸 ()* a) 𝑃𝐸 𝑡 + > 𝑃𝐸 ()* spontaneous conversionof 𝐴 → 𝐾 → 𝐼𝐸
FIG. 11 Schematic of release of available potential energy.(a) Motionless initial water glass with a non-zero interfaceslope between stably stratified water masses. (b) Motionlessfinal state after available potential energy A was released,converted to kinetic energy K , and then converted to internalenergy IE , which warms each water mass slightly. Lorenz’s APE theory assumes an initially stable verti-cal profile, so energy cannot be converted to work frompurely vertical rearrangement, and it requires the defini-tion of a reference state that minimizes the total poten-tial energy P subject to some constraints. Lorenz (1955,1967) proposed that the appropriate reference state is onein which the atmosphere is reversibly and adiabaticallyre-arranged (holding mass constant) to a static state ofminimum potential energy P r , leaving the residual po-tential energy available to drive motion. Such a refer-ence state is in mechanical equilibrium (motionless andin hydrostatic balance), but not in thermal equilibrium,given there remains a vertical temperature gradient asso-ciated with vertical stratification. The dry available po-tential energy is then defined over the entire atmosphere(Peixoto and Oort, 1992): A = (cid:90) Ω A ( P − P r ) ρdV. (66)Letting C ( X, Y ) indicate a rate of energy conversion fromreservoir X to reservoir Y , the Lorenz energy cycle is:˙ A = ˙ G − C ( A, K ) (67)˙ K = C ( A, K ) − ˙ D. (68)APE is generated at rate ˙ G and converted to kinetic en-ergy ˙ K . Kinetic energy is ultimately removed by fric-tional dissipation at rate ˙ D .The original APE of Lorenz (1955) was only definedfor a global integral over small-amplitude perturbationsfrom the resting state. The integrand in (67) may be lo-cally positive or negative, but the APE is positive definiteupon integration.Van Mieghem (1956) quickly pointed out limitationsin the assumptions of the reference state, remarking:4“the hydrostatic hypothesis and the assumption of in-compressibility which are commonly used in atmosphericdynamics are far more dangerous to introduce in energystudies”. Pearce (1978) formulated a more complete APEthat included the impact of energy available upon adia-batic rearrangement in the vertical direction and that isvalid without assuming that the atmosphere is in hydro-static balance.The need to define a positive-definite local APE (anAPE density) was addressed by Holliday and Mcintyre(1981) for a stratified, incompressible fluid. That sameyear, Andrews (1981) developed a theory for local APEdensity valid for nonhydrostatic and compressible flows.He identified an additional energy reservoir in compress-ible atmospheres termed the available elastic energy.Lorenz’s APE was shown to always be less than or equalto a volume integral of the APE density (Andrews, 1981).Shepherd (1993) derived a quantity equivalent to theAPE density using Hamiltonian methods and called it a“pseudoenergy” (we will briefly review Hamiltonian GFDin VIII.B).When the effects of moist processes are included, theestimated APE production rate is far greater than theobserved or estimated frictional dissipation rate (Pauluis,2007), which indicates that frictional dissipation ˙ D is notthe only (nor even primary) sink of moist APE. Pauluis(2007) formulated the first budget for APE that includessources, sinks, and diffusion of water content in a moistatmosphere. He showed that heat and water diffusion,precipitation and re-evaporation can each change APE.But despite the conceptual similarity between irreversibleentropy generation and APE destruction, there is not adirect mapping between the sign of the change in APEand the occurrence of irreversible moist processes; thesign of the APE change is also a function of the verticalposition of the air parcels in which these processes occur,relative to the vertical position of the same air parcels inthe adiabatically-rearranged minimum- P reference state.In a moist atmosphere, the vertical rearrangement of airparcels towards the reference state can be complex. Forexample, a moist parcel of air may be in a statically stableenvironment within the unsaturated boundary layer, butif it is lifted adiabatically until the water vapor begins tocondense and release latent heat, the parcel may never-theless acquire a higher altitude in the minimum- P ref-erence state, even if no horizontal gradients are present.Evaluating the APE involves applying a sorting algo-rithm to find the appropriate reference state, and thiscan be computationally intensive, particularly if moistureis considered (Hieronymus and Nycander, 2015; Lorenz,1979; Randall and Wang, 1992; Stansifer et al., 2017; Suand Ingersoll, 2016). Nevertheless, APE and the relatedLorenz energy cycle are widely applied in the atmospheric(usually using a dry formulation) and oceanic literature(see e.g., Hughes et al., 2009; Lembo et al., 2019; Li et al.,2007; von Storch et al., 2012). The zonal mean of APE has also been shown to scale with the eddy kinetic energyor ‘storminess’ of the midlatitude storm tracks (Schneiderand Walker, 2006), and this relationship has been usedto help explain future changes in storminess projected byglobal climate models (O’Gorman, 2010).
2. Exergetics
The Lorenz energy cycle is not directly translatableto the traditional study of thermodynamics. Lorenz’sand subsequent approaches that minimize potential en-ergy to define a reference state may be characterized asa ‘mechanical’ perspective (Huang and McElroy, 2015),in which the minimum potential energy reference state isnot necessarily a state to which the atmosphere sponta-neously tends. In contrast, the concept of exergy facili-tates a second-law based thermodynamic perspective onthe availability of energy to do work in a fluid whichhas been developed in parallel with APE theory (e.g.Bannon, 2005, 2012; Dutton, 1973; Dutton and Johnson,1967; Fortak, 1998; Huang and McElroy, 2015; Karlsson,1990; Keenan, 1951; Kucharski, 1997; Livezy and Dut-ton, 1976; Marquet, 1991, 1993; Marquet et al., 2020;Peng et al., 2015). Some physics curricula and most engi-neering programs teach the concept of exergy (coined byRant, 1956), which is used widely in the energy industry.In the climate literature it has also been called ‘staticentropic energy’ (Dutton, 1973), ‘static exergy’ (Karls-son, 1990), ‘availability’ (Bannon, 2013), ‘available en-ergy’ (Bannon, 2005, 2012), or ‘available enthalpy’ (Mar-quet, 1991), with small variations in formulation and as-sumptions. The subfield of study generally may be calledexergetics (Karlsson, 1990). The relationship betweenAPE and exergy of the atmosphere has been discussedin Dutton (1973); Fortak (1998); Kucharski (1997); Mar-quet (1991); Tailleux (2013).Exergy B is the amount of energy in an out-of-equilibrium system that can be converted to useful workupon moving to a reference state that is in thermody-namic equilibrium with its environment. The referencestate has lower total energy than the original state. Foran open system such as a power plant, we may imaginethe system exporting energy reversibly until it reachesthe reference state; the amount of energy exported isequal to the exergy.For the climate system, the definition of the “envi-ronment” is ambiguous, so the climate system is insteadassumed isolated from its star and deep space. An ex-ergy can be defined that measures the portion of energywithin the system that can perform work on the systemitself. As for the open-system case, the reference statehas lower total energy than the initial state, and the dif-ference is the exergy. But in the isolated-system case,the system cannot evolve to the reference state becauseit cannot export energy.5To define both APE and exergy, the reference state isreached assuming total mass is held constant. Howeverinstead of minimizing total potential energy, exergy is de-fined by minimizing the Gibbs free energy of the referencestate at a reference temperature T r . The reference stateis then both static (mechanical equilibrium) and isother-mal at T r (thermal equilibrium), with reference profiles inthe vertical of pressure p r ( z ) and specific entropy s r ( z ).The difference in the Gibbs free energy between the ref-erence state and the initial state is the exergy. The staticexergy B of a single fluid may therefore be defined (e.g.,Bannon, 2005; Fortak, 1998): B = h ( s, p ) − h ( s r , p r ) − α ( p − p r ) − T r ( s − s r )=[ h ( s, p ) − h ( s, p r ) − α ( p − p r )] (69)+[ h ( s, p r ) − h ( s r , p r ) − T r ( s − s r )]for enthalpy h = u + pα .The relationship between the energy of the referencestate and the exergy is sketched in Fig. 12; the solid blackcurve gives the entropy S as a function of total energy E for an isolated system in thermodynamic equilibrium.The initial out-of-equilibrium ( S, E ) state is at position a ,and the system’s total energy is fixed to lie along the line a − e . The lower-energy reference state can be identifiedas a position along the function S ( E ) bounded by points c and e . The function S ( E ) can be interpreted as thepartition between the reference state energy (to the leftof the curved line) and the remaining exergy B (to theright of the curved line), such that their sum equals thetotal energy. To be consistent with the definition of staticexergy above, we note that we neglect the small kineticenergy portion of the initial state’s total energy in thefigure.The reference temperature T r may be chosen such thatthe reference state can be reached reversibly, so thatentropy production δS = 0 (Karlsson, 1990), which isequivalent to selecting a T r to maximize exergy B = B max (position c in Fig 12). A reference state at lower T r can only be reached by reducing the total entropy ofthe isolated system, which violates the second law; a ref-erence state at higher T r cannot be reached reversibly(position d in Fig. 12), and the subsequent entropy pro-duction δS > B < B max .Exergy is the upper bound of work that can be extractedfrom a nonequilibrium system under purely reversibleprocesses; irreversible processes, including moist irre-versible processes that don’t perform work, act to destroyexergy. The second law of thermodynamics may there-fore be re-expressed in energy variables as dB/dt ≤ T r should not beconfused with the final equilibrium state that the system (initial state) 𝐸𝑆 𝐵max ac e 𝐵 𝛿 𝑆 ( 𝑇 ! ) bd (equilibrium state) FIG. 12 Total energy E vs. total entropy S . The solid blackcurve S ( E ) is the entropy of an isolated system of total en-ergy E in thermodynamic equilibrium. The initial state a andfinal equilibrium state e have the same total energy for an iso-lated system. The reference state c maximizes exergy B max if it can be reached isentropically (Karlsson, 1990). If T r ischosen such that the reference state cannot be reached isen-tropically, exergy available for mechanical work B < B max .Adapted from Landau and Lifshitz (1969) Fig. 2 and Huangand McElroy (2015) Fig. 5. would reach if isolated and allowed to evolve irreversibly(position e in Fig. 12). This equilibrium state is motion-less after all of the exergy has been dissipated to internalenergy, maximizing entropy, but maintaining the totalenergy of the system at its initial value.Similar to the decomposition of APE (Andrews, 1981),exergy of an ideal gas can be split into two contributions(Bannon, 2005, 2012; Karlsson, 1990; Marquet, 1991):an available elastic energy [first term in brackets of (69);zero in an incompressible atmosphere[, and an availablepotential energy [second term in brackets of (69)]. Ban-non (2012) pointed out that the exergy could also be ap-proximately partitioned into a component that could bereleased upon a stable resorting of each vertical column(the Available Convective Energy), and then upon com-pletion of that process, the remaining horizontal gradi-ents provide an Available Baroclinic Energy which drivesthe large scale horizontal flows. The difference betweenthe traditional APE approach and an exergetic one canbe reconciled if one considers the exergy within each layerof the atmosphere individually (Kucharski, 2001; Penget al., 2015).For moist atmospheres, one must also consider a chem-ical departure from a saturated reference state in order todefine exergy. This requires an additional additive term µ x ( q x − q x ) in (69) for each x th component, for somechemical potential µ per unit mass, and mass fraction6 q . Exergy correspondingly increases in a moist atmo-sphere due to an additional available chemical energy. Ingeneral, multicomponent atmospheres will need thermo-dynamic potentials and departures defined for each fluid(e.g. Bannon, 2005; Karlsson, 1990; Marquet, 1991). Be-cause the climate system exists in a gravitational poten-tial field, chemical potentials at equilibrium are functionsof the geopotential, such that µ x + g ♁ z =constant for gasspecies x (dry air and water vapor) , gravitational ac-celeration g ♁ , and altitude z (Hatsopoulos and Keenan,1965). One must also decide whether water mass is con-served in the atmosphere upon rearrangement to the ref-erence state (e.g., Livezy and Dutton, 1976; Marquet,1993) or whether the atmosphere is, more accurately, anopen system to water (Bannon, 2005; Pauluis, 2011).There is little consensus over how to choose the refer-ence temperature T r in exergy studies of the climate sys-tem (e.g., Bannon, 2012; Dutton, 1973; Karlsson, 1990)and it may be arbitrary (Kucharski, 1997). Generallyit is chosen to be around 250 K. Karlsson (1990) chosethe reference temperature as the one that minimized theentropy difference between the atmosphere and its equi-librium state, and Bannon (2013) showed that this wasequivalent to maximizing exergy. A difficulty arises be-cause there is no external thermostat setting the tem-perature toward which our climate may evolve. In ourprevious power plant example, it is reasonable to assumethat the plant is surrounded by a reservoir of roughlyconstant temperature; namely, the atmosphere. For theclimate as a whole, the only technically correct referencetemperature is the several-kelvin chill of deep space, andeven then only after the death of the sun. Instead, onecan imagine an Earth system that at some point suddenlybecomes isolated, no longer receiving nor losing heat andentropy to space . Under this hypothetical condition,entropy would increase toward a maximum value whiletotal energy would remain constant. What would the fi-nal state look like and how would an unforced atmospherefreely evolve toward it? It would certainly spin down dueto friction; it would become saturated due to contact witha frozen water surface but cloudless because of hydrome-teor fallout; and eventually, due to the very slow processof molecular heat transfer, the atmosphere would becomeisothermal (pressure and density would not homogenizebecause of the gravitational potential field). On evenlonger timescales, the various gases of the dry air mix-ture itself would begin to fractionate, as is hypothesized At equilibrium there are no condensed species aloft. This is a difficult thought experiment because radiation betweencomponents of the Earth system is so important (consider theatmospheric longwave warming of the surface). If the Earth sys-tem becomes thermodynamically isolated, can the componentsstill exchange energy radiatively? If so, how does one deal withthe radiative energy directed upward at the top of the atmo-sphere? to be happening on the giant planets.An exergetic budget provides an alternative way toformulate a mechanical efficiency of the climate sys-tem, which can be evaluated quantitatively using climatemodel output (Bannon, 2012; Karlsson, 1990). Karlsson(1990) defines a climate efficiency as ‘the global net con-version to kinetic exergy divided by the global net inflowof static exergy under the assumption of local thermo-dynamic equilibrium’, approximately C ( B, K ) / ˙ B . A dryexergetic analysis has been proposed by Lucarini (2009)and recently applied to Earth’s climate to study its sea-sonality (Huang and McElroy, 2015).Both the APE and exergetic frameworks provide a per-spective of irreversibility focused on energy rather thanentropy. The literature that explores exergy in the cli-mate system more closely appeals to fundamental ther-modynamical concepts. However the study of the exer-getics of the atmosphere is evidently hampered by theproposal of a unique terminology for nearly every paperpublished in the field [an early collection of which aretabulated in Marquet (1991) Appendix B]. It is likelythat a consensus, yet forthcoming, may make exerget-ics more palatable in the classroom and assist in defin-ing and quantifying useful measures of climate efficiency.Ideally this consensus will fall on terminology alreadywidely used in physics and engineering, so as to promotemore interdisciplinary collaboration and to avoid furtherconflation with the classical APE development of Lorenz(1955). B. Statistical mechanical approaches for steady flows
So far the discussion of atmospheric and oceanic circu-lation has emphasized its forced-dissipative nature andthe consequences of the system being heated at a highertemperature than that at which it cools. This can be con-ceptualized as the climate system being in contact withtwo thermal reservoirs of different temperatures, whichmakes thermodynamic equilibrium impossible. Energytransfer between the reservoirs occurs via overturningcirculations of various scales, from thunderstorms to theglobal atmospheric or oceanic circulation.Now we consider aspects of the atmosphere and oceanthat cannot be conceptualized as being in contact withtwo different thermal reservoirs. Such systems do not de-velop overturning circulations to move heat downgradi-ent, but they instead develop quasi-horizontal flows withcharacteristic organization and steadiness dominated byinertial forces relative to weak forcing and dissipation.Examples include eye-eyewall mixing dynamics in trop-ical cyclones (Schubert et al., 1999); mesoscale eddiesin the ocean (Venaille and Bouchet, 2011); the strato-spheric polar vortex (Prieto and Schubert, 2001); Rossbywave propagation at midlatitudes under a vorticity gradi-ent (Young, 1987); and Jupiter’s Great Red Spot (Miller7et al., 1992). Such coherent structures are ubiquitous inturbulent geophysical flows. Unlike the heat engine anal-ogy employed for the climate system, these phenomenacannot produce work; rather, they can be thought of asbeing thermodynamically isolated or in contact with asingle thermal reservoir, not two.The study of such fluid equilibria is a branch of geo-physical fluid dynamics (GFD) that has adapted equilib-rium statistical mechanics to the fluid equations: Hamil-tonian GFD. This framework applies Hamilton’s princi-ple of stationary action for conservative systems to con-tinuum fluid mechanics using statistical techniques validin the thermodynamic limit (infinite particles). Equilib-rium statistical mechanics has been used successfully todescribe a wide range of simple fluid behavior with ap-plicability to observed large-scale flows. More detailedtreatments may be found in reviews by Bouchet and Ve-naille (2012); Majda and Wang (2006); Morrison (1998);Salmon (1988); Shepherd (2003, 1990).
1. Theoretical development
Use of Hamilton’s principle of stationary action pro-vides an alternative way to derive the equations of mo-tion using the calculus of variations. However, the typi-cal differential approach using Newton’s second law easilyincorporates terms for friction, viscosity and other non-conservative forces, and this is a limitation on the ap-plicability of Hamiltonian formulations, which only existfor conservative systems. In spite of its idealism, Hamil-tonian geophysical fluid dynamics is an important andilluminating part of the literature on quasi-steady flows,and variational methods used in Hamiltonian GFD havehelped to explain the stability and longevity of some well-known enduring vortices. We’ll briefly survey four use-ful equilibrium statistical mechanics approaches and thenin section VIII.B.2 mention some interesting geophysicalapplications and successes.It might be surprising that forced-dissipative geophys-ical flows are amenable to variational methods. Ourocean and atmosphere are very high Reynolds number,strongly stratified fluids on a rotating planet. Due to ro-tation and stratification in particular, such fluids at largescales exhibit an approximately two-dimensional (2D),non-divergent flow field, which manifests as jets, waves,and vortices (e.g., Flierl, 1987). It is this 2D characterof large-scale geophysical flows that allows Hamiltonianmethods to be useful for their analysis.A non-divergent flow field is described by the inviscid,incompressible 2D Euler equation, d v h dt = − ρ ∇ h p (70)for horizontal flow field v h , where we continue to use d/dt as the material derivative. Energy conservation for this system is simply conservation of kinetic energy integratedover the domain. Mechanical work is limited to the innerproduct of a stress and a strain (Fang and Ouellette,2016). Two key aspects of (70) make the application ofHamiltonian methods feasible. Firstly, in 2D turbulence,kinetic energy is, on average, transferred to larger scales,avoiding the build up of kinetic energy at the smallestscales that would occur in 3D invisid flow. Secondly,taking the curl of (70) gives, dωdt = 0 , where ω = ∇ × v h is the vorticity. The vorticity is ma-terially conserved following fluid elements and conservedwhen integrated over the domain. Depending on the ge-ometry and boundary conditions of the system, the cir-culation (area integral of the vorticity) and in some caseseither zonal and angular momentum are also conserved following from Noether’s theorem that every symmetrycorresponds to a conservation law.Strongly stratified geophysical flows under the influ-ence of rotation are well approximated by a systemknown as the quasigeostrophic (QG) equations, whichshare the properties of the 2D Euler equations discussedabove. In the QG case, the relevant vorticity variableis known as the potential vorticity, and it includes a de-pendence on the stratification. The QG system is alsodependent on a length scale known as the Rossby ra-dius that does not appear in the 2D Euler equations butmay nevertheless be included in the statistical mechanicsframeworks we will discuss (e.g., DiBattista and Majda,2000; Salmon et al., 1976; Weichman, 2006).A remarkable property of ideal 2D continuum fluidsis that any domain integral that is only a continuousfunction of vorticity (or potential vorticity) is conserved.The most commonly considered functionals are integralsof vorticity to some power. The infinite invariants arecollectively referred to as the “generalized enstrophy in-tegrals” (Young, 1987), or Casimirs. The corresponding Common lateral boundary conditions in GFD are periodicboundaries in one or more dimensions. While these domainsconserve linear momentum in the periodic dimensions, they donot conserve angular momentum. This is easy to confirm witha simple thought experiment. Imagine you have a tropical cy-clone centered in the middle of your doubly-periodic domain andyou are positioned far in one corner of it—this corner, of course,being joined by the three other corners of the domain becausethere are no walls. Measure your angular momentum as a func-tion of your tangential speed and distance from the center of thedomain. Maintaining that radius, rotate some angle around thecyclone’s spin axis until you have crossed through the domainedge. You find yourself ‘re-entering’ the domain from the op-posite side. Though your angular momentum should not havechanged, you are now discontinuously closer to the storm cen-ter than you were before. Thus the domain lacks invariance torotation and cannot conserve angular momentum. ω be composed of a range ofvorticity levels σ , with a probability density ρ ( x , σ ) suchthat at each point in the domain, (cid:82) ρ ( x , σ ) dσ = 1. Thearea fraction of each vorticity level is conserved as theflow evolves. This results in a locally averaged vorticityfield ω ( x ) = (cid:82) ρ ( x , σ ) σdσ .9The Boltzmann mixing entropy of RMS theory is: S = − (cid:90) (cid:90) Ω ρ ( x , σ ) log ρ ( x , σ ) dAdσ. (71)Setting the first variation of this functional to zerosubject to the constraints of energy, circulation andthe Casimirs using Lagrange multipliers, one solves foran equilibrium probability distribution ρ eq ( x , σ ) thatcorresponds to the most-disordered macrostate—themacrostate corresponding to the largest number of mi-crostates. To ensure that the stationary point of S is amaximum, one must further check to see that the secondvariation is negative. The equilibrium probability distri-bution of vorticity may be related to the large-scale flowfield of the equilibrium state, and in principle it shouldrecover the long-time average of the circulation derivedfrom traditional time-dependent numerical models.Although there is no theoretical justification for the ne-glect of higher-order conserved Casimirs in an ideal fluid(Miller et al., 1992), DiBattista and Majda (2000); Ma-jda and Wang (2006); Majda and Holen (1997) arguedthat retaining higher order Casimirs is impractical andunnecessary when considering equilibria of quasi-steadyforced-dissipative geophysical flows. A real fluid has afinite number of atoms, so the idea of infinite invari-ants is not applicable. Turkington (1999) further pointedout that the RMS theory itself is effectively discretized,given the assumption that at some tiny limiting scale,vorticity is not smooth but rather constant and mixesergodically , and that this theoretical weakness is notdissimilar to the point vortex model of 2D flow (this hasbeen partially remedied by Bouchet and Corvellec, 2010).Turkington introduced an alternative approach to maxi-mizing entropy without having to conserve the Casimirs.This theory conserves only energy, circulation, and themean and extrema of vorticity-like variables, and it isbased on an understanding of entropy from informationtheory (Jaynes, 1957a,b).The minimum enstrophy approach to HamiltonianGFD embodied in the selective decay principle has beencompared favorably to maximum entropy approaches ofvarious forms by Brands et al. (1999); Chavanis and Som-meria (1996); Huang and Driscoll (1994); Majda andHolen (1997); Naso et al. (2010); Schubert et al. (1999).Naso et al. (2010) showed an equivalence between thestatistical equilibrium state of a truncated fluid systemthat only conserved energy, circulation and enstrophyfor which the entropy is maximized, and the equilibriumstate solved for when enstrophy is minimized. Minimumenstrophy theory continues to be used to make predic-tions of steady state flows (e.g., Conti and Badin, 2020; All equilibrium statistical mechanics development requires an as-sumption of ergodicity, which remains unproven and doesn’t al-ways hold up well in experimentation (e.g., Brands et al., 1999).
Naso et al., 2011). In fact there remains no clear con-sensus in the literature about a superior formulation ofentropy-maximizing equilibrium methods, in large partbecause successful applications are so domain- and scale-specific.
2. Applications
Turning back to the observable flows that motivatedmuch of the theoretical development, we briefly highlighta few applications to demonstrate the usefulness of sta-tistical mechanics in understanding large scale flows inthe inertial limit. In particular, we will discuss four suchexamples: O (100 km) oceanic rings, the stratosphericpolar vortex, tropical cyclone eye-eyewall dynamics, andEarth-sized vortices on Jupiter. Differential planetaryrotation, stable stratification and column stretching (in-cluding that due to topography, e.g., Salmon et al., 1976)are essential aspects of real quasi-steady coherent struc-tures.The ocean is full of coherent, relatively long-lived west-ward propagating vortices (Chelton et al., 2007, and seean example in Fig. 13a) These vortices are surface-intensified, so they are not strongly damped by the fric-tional ocean floor. Venaille and Bouchet (2011) used a1.5 layer quasigeostrophic model of the ocean and usedRMS theory to identify equilibrium flows in the inertiallimit. They found that oceanic vortices (rings) can beconsidered as local equilibrium states, and their differ-ential latitudinal drift (poleward for cyclones and equa-torward for anticyclones) could be interpreted as an evo-lution toward potential vorticity homogenization. TheRMS equilibrium states showed some similarity to ob-servations, suggesting these flows are weakly forced anddissipated.David et al. (2018) recently applied RMS theory to aforced-dissipative numerical ocean simulation and foundthat the maximum entropy principle, using only a trun-cated conservation of Casimirs, was able to account forsome behavior of the nonequilibrium system. They sug-gest that statistical mechanics be reconsidered as a viablemethod of improving model parameterizations.In the winter hemisphere, a stratospheric polar vor-tex (the “polar night jet”) sets up and generally actsas a barrier to the mixing of chemical constituents be-tween the polar cap and the midlatitudes. This is par-ticularly consequential in the Southern Hemisphere be-cause it maintains a region of extremely low ozone, con-tributing to the existence of the ozone hole in Australspring (Fig. 13b, a low-mixing regime). In the NorthernHemisphere, there are regular polar vortex breakdowns(“stratospheric sudden warmings”) during which the coldpolar cap, previously maintained right on the pole andprevented from mixing by the jet-like edge of the polarvortex, suddenly experiences barotropic instability and0 a bc d FIG. 13 Examples of geophysical flow phenomena that have been described using equilibrium statistical mechanics. (a)Streaklines of ocean surface flow exhibiting oceanic rings from a numerical model. Image credit: NASA/Goddard SpaceFlight Center Image Visualization Studio; (b) Southern Hemisphere ozone hole on October 12, 2018, indicating location ofstratospheric polar vortex. Image credit: NASA Earth Observatory, NASA/NOAA Suomi NPP satellite; (c) Submesoscalevortices mixing the eyewall of Hurricane Isabel on September 12, 2003. Image credit: Defense Meteorological Satellite Program(DMSP) image of Hurricane Isabel at 1315 UTC 12 Sep 2003 from Kossin and Schubert (2004); (d) Jupiter and the Great RedSpot as seen by the Hubble Space Telescope on August 25, 2020. Image credit: NASA, ESA, STScI, A. Simon (Goddard SpaceFlight Center), M.H. Wong (University of California, Berkeley), and the OPAL team. mixes rapidly with warmer equatorward air via nonlin-ear wavebreaking (a high-mixing regime). It is of inter-est to see whether these two different regimes of polarvortex behavior can be captured by equilibrium statisti-cal mechanics. Prieto and Schubert (2001) tested bothmaximum entropy theory and minimum enstrophy the-ory to determine which approach could more accuratelypredict a zonally symmetric equilibrium state for eachregime compared to direct numerical integration of ide-alized initial vortices with passive tracers. The minimumenstrophy prediction was superior for the case where mix-ing occurred only within the polar cap. The more vi-olent scenario of domain-wide mixing, reminiscent of astratospheric sudden warming, was better predicted bythe maximum entropy solution.In general, statistical mechanical techniques are lim-ited to flows that are weakly forced and damped, andthey don’t apply to transient tropospheric weather sys-tems. One of the most speculative geophysical applica- tions of equilibrium statistical mechanics has been at-tempted at a very small scale in both space and time.Tropical cyclones exhibit the potential for barotropicinstabilities in the eye/eyewall region, which results inhighly asymmetric vorticity mixing (Fig. 13c). Schubertet al. (1999) numerically integrated the evolution of anidealized TC eyewall vorticity ring using the 2D Eulerequation. Barotropically unstable initial conditions ledto similar polygonal mixing patterns. They comparedthe numerical well-mixed end state to solutions reachedusing both the minimum enstrophy theory and maximumentropy theory. Overall the maximum entropy theorypredicted the numerical final state better. Whether theequilibrium state has any relevance to the actual evolu-tion of a tropical cyclone is less clear, because such stormsare very dynamic and transient (and very expensive toobserve in situ ).The Solar System’s most famous vortex does not hap-pen to be on Earth. The Great Red Spot of Jupiter1(Fig. 13d) has long been a source of inspiration for equi-librium statistical mechanics (Miller et al., 1992). Nu-merical integration of an annulus to which an externalpotential is applied, representing Jupiter’s rapid rotationand zonal shear flow, demonstrated that a single large-scale coherent vortex can uniquely survive for many eddyturnover times in such an environment (Marcus, 1988)and experimentation concurred (Sommeria et al., 1988).Miller et al. (1992) compared the equilibrium solutionsof RMS theory to the numerical experiments of Marcus(1988) and found good qualitative agreement. Turking-ton et al. (2001) were able to retrieve realistic equilibriaof Jovian anticyclones using the constrained theory ofTurkington (1999) if the initialized vorticity distributionskewed anticyclonically. Bouchet and Sommeria (2002)extended RMS theory to include quasigeostrophic flowsin the limit of small deformation radius, and found aGreat Red Spot-like vortex as a maximum entropy equi-librium structure. Under only slightly different param-eters the vortex was absent, suggesting an explanationfor the lack of such a vortex observed in Jupiter’s North-ern Hemisphere (which has a different zonal jet structurethan the Southern Hemisphere).A new vortical puzzle has just been furnished byNASA’s Juno mission to Jupiter, which captured imagesof the polar caps for the first time to reveal crystalline-like polygonal arrangements of cyclones centered on eachpole (Fig. 14a,b; Adriani et al., 2018). The polygo-nal arrangements appear remarkably steady and may bethe first geophysical observation of the strictly 2D vortexcrystal phenomenon (Adriani et al., 2018; Grassi et al.,2018; Tabataba-Vakili et al., 2020), which was first iden-tified by Fine et al. (1995) experimentally in a 2D elec-tron plasma (Fig. 14c). The potential of such remark-able plasma structures to have application to geophysicalflows was noted by Schubert et al. (1999).Using numerical simulation and scaling theories, bothshallow (O’Neill et al., 2015, 2016) and deep models (Gar-cia et al., 2020) suggest a range of statistically steadyvortex-dominated behavior on giant planet caps. Suchwork is able to explain Saturn’s isolated, steady polarcyclones. However, to date, no theory or modelling hasbeen able to achieve a steady crystalline structure of po-lar vortices from random initial conditions as observedon Jupiter, though some recent progress has been madein achieving such states transiently (Li et al., 2020). Thisis a new area of research that is ripe for an equilibriumstatistical mechanics study.Robert and Sommeria (1991) suggest that equilibriumstatistical mechanical techniques could potentially beapplied to a changing climate, given the separation oftimescales between the rapid statistical adjustment offlows compared to the slowly varying Lagrange multipli-ers that represent the forced climate response. DiBattistaet al. (2001) showed that equilibrium statistical mechan-ics can predict ‘meta-stable’ states of systems that are secularly driven, and the most likely state varied withthe forcing dynamically. Other work shows the viabil-ity of second-law based equilibrium tools for turbulentforced-damped flows (e.g., David et al., 2018). Thereis likely further opportunity to bridge the statistical andthermodynamical interpretations of entropy to better un-derstand complex, out-of-equilibrium flows on Earth andother planets, given the domain-specific success of eachapproach.
C. Maximum entropy production principle controversy
While statistical mechanics is well developed for sys-tems in equilibrium, it is vastly less developed and ac-cepted for systems out of equilibrium like the Earth sys-tem. Knowing the most probable end-state of a systemdoes not necessarily yield information about the paththat a system will take to get there other than that itmust satisfy the second law. However, much theoreticaland empirical work has sought a variational principle thatcould identify a preferential path that a system travelstoward equilibrium in simple fluid systems, and for lin-ear systems very close to, but not in, equilibrium (e.g.,Busse, 1967; Jaynes, 1980; Malkus and Chandrasekhar,1954; Malkus and Veronis, 1958; Palm, 1972; Prigogine,1962).A more wide-ranging variational hypothesis was pro-posed by Paltridge (1975, 1978, 1979) with applicationto the out-of-equilibrium steady state of climate. This“maximum entropy production principle” (MEP princi-ple) hypothesizes that the climate system adjusts it-self to be in a state that maximizes entropy production[see reviews by (Kleidon and Lorenz, 2005; Martyushevand Seleznev, 2006; O’Brien and Stephens, 1995; Ozawaet al., 2003)]. The MEP principle is a controversial hy-pothesis because it is not derivable from the equationsof motion. None of the tools of the previous sections areclearly applicable here.Paltridge (1975) considered a highly idealized climatemodel where both the top of atmosphere and planetarysurface are in energy balance. The hypothesis that thenonequilibrium climate system has sufficient degrees offreedom to be “controlled by some minimum principle”was proposed and tested. Paltridge constructed a ten-box climate model of the Earth spanning pole-to-polewith three free parameters: cloud areal fraction, temper-ature and the sum of latent and sensible heat flux at thesurface. Each box contained two energy balance equa-tions; one for total energy balance (top-of-atmosphere) This should not be confused with the maximum entropy produc-tion principle established in statistical mechanics for the relax-ation of simple systems toward equilibrium (e.g., Robert, 2003;Robert and Sommeria, 1992), nor the related ‘maximum caliber’hypothesis (Ghosh et al., 2020; Press´e et al., 2013). FIG. 14 Jupiter’s (a) North and (b) South Polar Cyclones imaged by the NASA Juno mission in infrared (Adriani et al., 2018).(c) Vortex crystal experimental equilibria (Fine et al., 1995). and one for oceanic energy balance. Lacking a third bal-ance equation to solve for the three free parameters ineach box, Paltridge showed that minimization of entropyexchange with the environment (min( dS e /dt )) betweenthe ten boxes yielded realistic values of meridional heatflux. The only other constraint to the system was con-servation of total energy. This entropy exchange mini-mization closed the system and was able to produce aprediction of surface temperature and cloud cover with“extraordinary accuracy” .Rodgers (1976) argued that the simplicity of the model(lack of physics) suggested it should be broadly appli-cable to other planets but would clearly fail if appliedto Mercury or Mars. Rodgers also pointed out that, insteady state, dS/dt = 0, and minimizing dS e /dt is equiv-alent to maximizing internal entropy production. Pal-tridge (1978) adopted this interpretation and extendedthe ten-box model to a two-dimensional global climatemodel with 380 cross-box energy flows. Upon minimizingentropy exchange (maximizing entropy production), theresults were a surprisingly good fit to global distributionsof temperature and cloud cover. His results suggestedthat the MEP principle was able to capture large-scaleobserved circulation patterns. However, Mobbs (1982)noted that this is likely due to the model’s albedo dis-tribution being fixed to the observed albedo distribu-tion. Even with this observational constraint, the two- This work and most subsequent studies of the MEP principleadopt a material view of the climate system; entropy productionfrom the absorption of ordered shortwave radiation and emissionof disorderly longwave radiation are neglected as discussed insection II.A.1 (Pujol and Llebot, 1999). dimensional ocean energy transport was quite different tothat observed in some regions (Sohn and Smith, 1994).Interest in the potential for MEP to provide closurefor underconstrained climate problems prompted furtherdevelopment and application to energy balance models(e.g., Gerard et al., 1990; Grassl, 1981; Lorenz et al.,2001; Mobbs, 1982; Nicolis and Nicolis, 1980; Sohn andSmith, 1993, 1994). While such studies reported somesuccessful applications, in general the results were mixed.Moreover, there are a number of difficulties with the MEPprinciple that, in our view, limit its applicability: a. The MEP principle is incomplete.
Regarding a two-boxmodel MEP application (e.g., Lorenz et al., 2001; Pal-tridge, 2001), a re-examination demonstrates that theMEP state is neither inevitable nor mathematically jus-tified (Nicolis and Nicolis, 2010). Mobbs (1982) foundin a simple model that a finding of a maximum entropyproduction state or a minimum entropy production statecould each occur, depending on the parameterization ofthe diffusivity in the model. Goody (2007) pointed outthat the MEP principle was unconstrained, meaning thataside from mass and energy conservation, there was noway to consider external factors of a planet like its sizeor rotation rate, which are among demonstrably impor-tant constraints on the general circulation. Further con-straints eventually obviate the need for a variational prin-ciple entirely (e.g., Chang, 2019; Goody, 2007). b. The MEP principle appears to fail when applied to ver-tical transport.
Early applications of the MEP princi-ple applied it to vertically-integrated representations of3the climate system, for which only horizontal entropy ex-changes are present (e.g. Lorenz et al., 2001). However,the vast majority of the entropy production on Earth isassociated with vertical entropy fluxes (Lucarini et al.,2011). When MEP is applied to vertical entropy ex-change, it often gives unphysical predictions; for example,temperature distributions that are gravitationally unsta-ble or atmospheric layers that destroy entropy (Chang,2019; Herbert et al., 2013; Ozawa and Ohmura, 1997;Pujol and Fort, 2002). As above, this suggests a needto include additional constraints to the MEP framework.Lucarini et al. (2011) argued that a model with no ver-tical resolution is not able to test the MEP hypothesis,given the dominance of irreversible entropy productiondue to vertical motion. The conceptual model of theatmosphere/climate system as a heat engine as articu-lated by Lorenz (1955, 1967) includes a substantial rolefor convection, which is neglected in climate models thatlack vertical resolution. Pointing out that MEP as formu-lated doesn’t know about (for example) the gravitationalpotential of a planet’s atmosphere, Chang (2019) devel-oped a constrained MEP principle to seek meridional fluxsolutions only within the bounds of a stable vertical strat-ification. The author found that sufficient constraints toensure physicality no longer required a variational clo-sure. c. The MEP principle cannot account for previous climatestates or future warming of Earth.
Grassl (1981) demon-strated that Paltridge’s procedure predicted negligiblepolar warming under a doubling of CO , because it lackedan ice-albedo feedback. Gerard et al. (1990) suggestedthat their MEP findings are consistent with relativelysteady global temperature across deep time, and incon-sistent with glaciation periods. Paltridge et al. (2007)attempted to apply the MEP principle to a GCM witha water vapor feedback and found an implausible reduc-tion of cloud cover upon a doubling of CO concentra-tion (Paltridge, 2009). Paltridge et al. (2007) also up-dated and “re-tuned” the original Paltridge (1975) en-ergy balance model, now with an inclusion of a constraintthat maximized vertical heat flux from the ground to theatmosphere, thus maximizing total entropy productionfrom both vertical and horizontal energy transport. Thisattempt failed at reproducing realistic temperatures andareal cloud fractions. d. The specification of the MEP optimization problem is am-biguous. The MEP principle has also been tested inglobal climate models (e.g., Fraedrich and Lunkeit, 2008;Ito and Kleidon, 2005; Kleidon et al., 2006; Pascale et al.,2013). Kleidon et al. (2003) considered a simplified globalclimate model with no moisture or radiation in which theonly irreversible entropy production occurs through fric- tional dissipation. The authors argued that MEP couldbe used to determine the “correct” surface friction pa-rameters in the model. But Chang (2019) pointed outthat using the irreversible entropy production in the at-mosphere to set the surface friction parameters implicitlysets land/ocean areal fraction and roughness as primarilyfunctions of the atmospheric dynamics, which is implau-sible. Pascale et al. (2012a) ran an experiment similarto Kleidon et al. (2003) but varied the convective en-trainment rate among other parameters, finding that theresulting climate failed to maximize entropy productionfor parameter combinations that produce the most real-istic climate. They demonstrated that entropy produc-tion increased monotonically with absolute temperature,partly because of the dominance of the hydrological cy-cle in the entropy budget. Instead, Pascale et al. (2012a)did find some evidence for the Lorenz (1960) claim thatthe climate state is one that maximizes APE generation.Which aspects of a given representation of the climateshould be adjusted to obtain the MEP state clearly mustbe agreed upon before this principle can be accepted asone that governs the climate. e. The MEP principle lacks a sound physical basis.
In spiteof several efforts (e.g., Paltridge, 1979, 2001), no theoret-ical justification had been found for MEP, rendering itsacceptance among climate scientists and physicists quitelimited. Dewar (2003, 2005) attempted to provide a theo-retical basis for the MEP principle using the Shannon en-tropy from information theory (Jaynes, 1957a,b), whichrepresents an interesting bridge between the thermody-namic entropy measured by climate scientists and the sta-tistical mechanical entropy considered by fluid dynami-cists. This development was severely challenged with thepublication of technical rebuttals to the Dewar papers(Bruers, 2007; Grinstein and Linsker, 2007). Grinsteinand Linsker (2007) argued that key assumptions in thederivation of Dewar (2005) required the system in ques-tion to be very close to equilibrium, which Dewar (2009)conceded.Despite the difficulties described above, MEP researchis ongoing, and proponents argue that it has applicationsas wide ranging as the evolution of river networks, eco-nomics, biotic activity and the Gaia hypothesis (Kleidonand Lorenz, 2005). More recent work recasts MEP asan “inference algorithm” rather than a physical princi-ple, and claims that it is effectively not falsifiable (De-war, 2009). For example, Dyke and Kleidon (2010) state:“any empirical evidence that does not agree with a MEPprinciple prediction does not represent any falsificationof the theory, rather the identification of the ’fact’ thatthe system is not an MEP system”.The MEP principle is understandably appealing be-cause it allows for a solution of the steady state of theEarth’s climate without solving for its complicated dy-4namical evolution. Yet the principle lacks theoreticaljustification as well as consistent numerical and obser-vational success. We conclude that the validity and use-fulness of the MEP principle remain aspirational.
IX. CONCLUSIONS & PERSPECTIVES
We have reviewed the key scientific developments inthe application of the second law of thermodynamics tothe climate system. The climate system may be definedin a number of ways, each of which differs in the extentto which radiation is treated as part of the system ratherthan as part of the surroundings (Bannon, 2015; Gibbinsand Haigh, 2020). By focusing on exclusively on matterwithin the climate system, the second law may be seen toprovide a direct constraint on the rate at which the cli-mate system’s heat engine performs work (Goody, 2003;Pauluis and Held, 2002a).The heat-engine perspective on atmospheric circula-tions was shown to allow for theoretical constraints onconvective updraft velocities (Emanuel and Bister, 1996),tropical cyclone intensity (Emanuel, 1986), and the at-mospheric meridional heat transport (Barry et al., 2002).While the theory of potential intensity of tropical cy-clones has been quite successful (Emanuel, 2018), afirst principles theory for convective updraft velocities(Pauluis and Held, 2002a; Singh and O’Gorman, 2016) re-mains elusive, although ongoing work is promising (Par-odi and Emanuel, 2009; Seeley and Romps, 2015; Singhand O’Gorman, 2013, 2015). Theories for the merid-ional heat transport by the atmosphere are limited bythe extent to which they account for the effects of moistprocesses on the atmospheric circulation, which is par-ticularly influential in governing the response of the at-mospheric circulation to global climate change (Lalibert´eet al., 2015; Singh and O’Gorman, 2016).Interrogating the entropy budget and heat-engine char-acteristics of climate models and of other planets mayprovide a pathway toward better understanding of theEarth’s climate system (Lembo et al., 2019; Lucariniet al., 2010a). The heat engines of other planets differ infundamental ways from that of the Earth, challenging ourimplicit assumptions about the thermodynamics of plan-etary circulations (e.g., Koll and Komacek, 2018). Globalclimate models are rarely developed with the second lawin mind; entropy-based diagnostics therefore provide op-portunities for model evaluation (Lembo et al., 2019).However, challenges remain in accurately modeling thesecond law in global climate models because their entropyproduction occurs in complex subgrid scale parameteri-zations which represent both reversible and irreversibleprocesses (e.g., Gassmann and Blender, 2019). The rapidincrease in computing power each decade continues to re-duce the need for such subgrid scale parameterizations assmaller scales are explicitly resolved. But it will be many decades to centuries before direct numerical simulation ofthe atmosphere and ocean at Reynolds numbers charac-teristic of geophysical flows is possible.We have also described the application of variationalmethods to the climate system. We discussed states ofminimum free energy (Bannon, 2005) and minimum po-tential energy (Lorenz, 1955) as methods for determiningthe atmosphere’s ability to perform work. A statistical-mechanics formulation for two-dimensional geophysicalflows was also described in which concepts such as en-tropy are used in a completely different context (Bouchetand Venaille, 2012). Nevertheless, the idea of irreversibil-ity is key to this formulation, and it may be described asan application of the second law to the climate system.Finally, we discussed the maximum entropy produc-tion (MEP) principle proposed by Paltridge (1975). Al-though we argue that a sufficient basis for accepting theMEP principle has not been established, we note thatit has motivated a great deal of research into Earth’sentropy budget (e.g., Goody, 2003; Pascale et al., 2011;Peixoto et al., 1991; Stephens and O’Brien, 1993), whichhas no doubt contributed to an improved understandingof irreversible processes within the climate system.This review has also highlighted a number of promis-ing directions for future research. In particular, we haveshown that global climate models and higher-resolutionmodels run in RCE appear to disagree as to whether thework performed by the atmospheric heat engine will in-crease or decrease in a warmer climate (Lalibert´e et al.,2015; Singh and O’Gorman, 2016). Further analysis ofthe atmospheric heat engine in both types of modelscould shed light on this important question (Lembo et al.,2019; Lucarini et al., 2010a). Further, we presented newanalysis of the effect of convective organization on themechanical efficiency of moist convection. To our knowl-edge, such an analysis has not been presented previously,but it may have important implications for our under-standing of convective organization, particularly in thecontext of global warming (Wing and Emanuel, 2014).The range of research in classical thermodynamicsand statistical mechanics that can be brought to bearon climate questions is broad and in many cases well-developed. Indeed, statistical mechanics approaches havealready been applied to climate models in the form ofapplications of Hamiltonian fluid mechanics to the prob-lem of parameterization of ocean eddies (David et al.,2018) and the use of statistical mechanics principles forstochastic parameterization as recently reviewed by Ghiland Lucarini (2020). Building on these examples requiresincreased collaboration between climate scientists andphysicists which we hope this review will help to foster.5
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