## 1. Introduction

*E*of the system is the sum of the rate of heating

*S*of the system is the sum of the entropy change rates due to the heating and its internal, irreversible entropy production

The idea that irreversible entropy production is a valid tool to estimate the climate efficiency is not without precedent. Brunt (1939) compares the eddy dissipation of kinetic energy with the incoming solar radiation to obtain an estimate of 2% for the conversion of the radiation to kinetic energy. Wulf and Davis (1952) formally incorporate Brunt’s dissipation in an entropy approach of the efficiency. Lorenz (1967) makes estimates similar to Brunt’s for the general circulation. In addition to frictional dissipation, Pauluis and Held (2002a,b) emphasize the entropy production due to the diffusion of heat, the diffusion of water vapor, the irreversibility of phase changes, and the frictional dissipation resulting from falling precipitation.

All these dissipative processes produce entropy, and that entropy production, always positive, is a measure of the internal activity of the climate system. The heat-engine relation in (1.4) provides the structure to formally define an efficiency associated with these entropy-producing processes. In particular, this structure requires determination of the heating input and the output temperature. The latter differs from the temperature, say, of where the frictional dissipation occurs.

The purpose of this study is to quantify the material entropy production efficiency (1.5) of Earth’s climate system. It is shown that the results are dependent on the definition of the climate system. Two different representations of the system appear relevant. The first representation defines the climate system as the sum of its material components, including the atmosphere, ocean, land, and cryosphere. This material system constantly exchanges energy with the surrounding radiation field. The second representation uses the engineering concept of a control volume (e.g., Bejan 2006) and defines the climate system as the sum of the material system and the radiation contained within its volume. The control volume exchanges energy and entropy only through the top-of-the-atmosphere (TOA) radiation fluxes. These two different representations provide complementary insights into the extent of the entropy production and, hence, into the efficiency of the climate system. Goody (2000, section 2) has discussed the difference in the entropy balance equation for the two representations and suggests that the material system is preferable. Nonetheless, neither representation is more fundamental, and both are examined here.

These two representations of the climate system clearly differentiate between the system and its surroundings. In contrast, the climate efficiency analyses of Johnson (1989, 2000) and Lucarini (2009) do not clearly distinguish between the climate system and its surroundings. These analyses include internal heating by sensible and latent heat fluxes with the radiative heating as part of the external heating of the climate system by its surroundings. They further subdivide the material climate system into regions of mean warming and cooling by all diabatic processes. As a consequence, the interaction between the system and its surroundings is improperly handled in their formulations of climate efficiency. [It is noted that this shortcoming does not invalidate the necessity of climate models to simulate the system’s entropics accurately (Johnson 1997).] The present approach treats the interaction properly.

Section 2 presents the energetics and entropics of the material representation of the climate system for which the radiation field constitutes the surroundings. The heat-engine relation for the material system is derived, and it is shown that the system does no external work. Thus, the productivity of the system is measured solely by its irreversible entropy production. The analysis accurately includes nonequilibrium thermodynamics (de Groot and Mazur 1984). Section 3 presents the energetics and entropics of the time-dependent radiation subsystem following Essex (1984) and Callies and Herbert (1988). Section 4 derives the heat-engine relation for the control-volume representation of the climate system, combining both the material and radiation subsystems below the top of the atmosphere. Section 5 applies both the material-system and control-volume representations to an idealized model climate system in radiative–convective equilibrium. Such radiative–convective models are useful tools in climate studies (e.g., Ramanathan and Coakley 1978), but their entropy aspects are rarely examined. The single-slab atmosphere model used here illustrates the various abstract aspects of the entropy production, particularly those associated with the irreversible radiation entropy production. The bulk atmospheric budgets found here are broadly consistent with the multilevel radiative–convective models of Li et al. (1994) and Wu and Liu (2010b) that do examine radiation entropics. Section 6 presents the conclusions and estimates the efficiency of Earth’s climate system using the two representations.

## 2. The material climate system

*e*is the sum of the specific kinetic, internal, and gravitational energies:

*p*is the pressure,

**v**is the mass-weighted, barycentric velocity, and

*V*of the climate system yields the statement of the first law of thermodynamics:

*s*satisfies

*k*th component of the multicomponent system. The summation convention is assumed for repeated indices. The rate of irreversible material entropy production per unit volume

*k*th component, and

*k*th component. Integration of (2.4) over the volume of the system yields, with

## 3. The radiation system

**m**is the unit vector normal to the surface element through which the radiation passes. Because energy is conserved between the material and radiation systems during emission and absorption, the material heating rate per unit volume

*T*of the matter. The conservative scattering makes no direct contribution in (3.1) because the scattering only changes the direction of the photons. The integral of the radiation energy density equation (3.1) over the volume of the climate system yields an equation for the radiation energy

## 4. The control-volume climate system

*T*of the body. In nonequilibrium conditions, the net effect of the interaction of the matter and the radiation is irreversible, and the entropy of the combined system increases. This irreversibility is analogous to that between any two systems that exchange heat (e.g., Kondepudi and Prigogine 1998, section 3.5). The explanation of this irreversibility is straightforward. Consider two systems (systems 1 and 2) with temperatures

*dQ*> 0 from system 2 to 1, the entropy of system 1 increases by

*T*. Thus, it is physically incorrect to describe (4.4) as an irreversible change solely to the radiation entropy, as done in Goody and Abdou (1996). It results from the entropics of both the radiation and the matter. This argument is further illustrated for the model climate system discussed in section 5.

The general result (4.10) is referred to as control-volume case 1 (CV1). Two additional cases of the control-volume representation are possible. Both cases remove the scattering processes from consideration by taking the incoming radiation to be the absorbed solar radiation. Then the fluxes transform as follows:

An exergy analysis (e.g., Bejan 2006) follows by multiplying the entropy equation (4.7) by a constant reference temperature

## 5. A radiative–convective equilibrium model of the climate system

An idealized model of the climate system in radiative–convective equilibrium illustrates the effect of the representation of the climate system on the Carnot efficiency and on the material entropy production efficiency *α* of the incoming solar radiation. For simplicity, the atmosphere is assumed to be nonreflecting. The atmosphere absorbs a fraction

A radiative–convective equilibrium model of the climate system consists of an isothermal atmosphere at temperature

Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0361.1

A radiative–convective equilibrium model of the climate system consists of an isothermal atmosphere at temperature

Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0361.1

A radiative–convective equilibrium model of the climate system consists of an isothermal atmosphere at temperature

Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0361.1

The incoming solar energy and entropy fluxes for the model are _{sun} = 6.77 × 10^{−5} sr (Wu and Liu 2010a), with ^{−8} W m^{−2} K^{−4}. For these settings, ^{−6}, and ^{−6}.

The terrestrial energy and entropy fluxes are those for a gray-/blackbody. The blackbody surface emissions are

In addition to radiative exchange, the model surface and atmosphere exchange energy through a convective–conductive heat flux *F*_{CHF}. By convention, *F*_{CHF} is positive if energy is transported from the warm surface to the cooler atmosphere. The convective heat flux is assumed to be a fraction *α*, *β*, *γ*, and *ε*. The parameter settings are

### a. Model energetics

*T*

_{p}= 255 K. Results for the temperatures and fluxes, summarized in Table 1, indicate representative surface and atmosphere temperatures of 278 and 253 K, respectively. Figure 3 summarizes the energetics of the model.

Temperature *T*, flux of energy *F*, and flux of entropy *J* for the radiative–convective equilibrium model of the climate system (Fig. 2). The radiation fields include the incoming, scattered, and absorbed solar radiation (ISR, SSR, and ASR, respectively), as well as the outgoing longwave radiation (OLR). The material fields include the surface, atmosphere, and convective heat flux (SFC, ATM, and CHF, respectively).

Energy budget (W m^{−2}) of the radiative–convective equilibrium model. Dashed arrows represent energy fluxes of solar radiation, solid arrows represent fluxes of terrestrial radiation, and the open arrow represents the convective heat flux. Numbers in the atmosphere and below the surface represent the net gain or loss of energy due to heating by the solar, terrestrial, and convective fluxes. The net heating in the atmosphere and surface is zero. The sum of the TOA fluxes is also zero.

Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0361.1

Energy budget (W m^{−2}) of the radiative–convective equilibrium model. Dashed arrows represent energy fluxes of solar radiation, solid arrows represent fluxes of terrestrial radiation, and the open arrow represents the convective heat flux. Numbers in the atmosphere and below the surface represent the net gain or loss of energy due to heating by the solar, terrestrial, and convective fluxes. The net heating in the atmosphere and surface is zero. The sum of the TOA fluxes is also zero.

Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0361.1

Energy budget (W m^{−2}) of the radiative–convective equilibrium model. Dashed arrows represent energy fluxes of solar radiation, solid arrows represent fluxes of terrestrial radiation, and the open arrow represents the convective heat flux. Numbers in the atmosphere and below the surface represent the net gain or loss of energy due to heating by the solar, terrestrial, and convective fluxes. The net heating in the atmosphere and surface is zero. The sum of the TOA fluxes is also zero.

Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0361.1

### b. Model entropics

^{−2}K

^{−1}is consistent with Wu and Liu (2010a). The entropy production of the atmosphere and the surface can be found in a manner similar to that in Planck (1959, section 102). The total entropy production associated with the interaction of the matter with the radiation fields is the sum of the production of entropy in the matter due to heating plus the entropy produced by the emission of radiation less the entropy extracted from the radiation field by absorption. For the atmosphere, we have

^{−2}K

^{−1}is dominated by the increase in radiation entropy due to terrestrial emission, both upward and downward, less the absorption of solar radiation and terrestrial radiation emitted by the surface. Similarly, the surface entropy production of 505 mW m

^{−2}K

^{−1}is dominated by the increase in radiation by terrestrial emission and solar scattering less the absorption of downwelling terrestrial radiation from the atmosphere and of solar radiation. The sum of (5.5) and (5.6) yields the net entropy production by the climate system. This net production agrees with the net outgoing flux of entropy in (5.4). We find

*J*

_{SSR}= 110 mW m

^{−2}K

^{−1}) is assigned to the surface entropy budget. We note that dividing the scattering processes between the surface and atmosphere would redistribute the scattering production between the two. Such redistribution would affect the local, but not the global, entropics of the model. Figure 4 summarizes the Planckian entropics of the model.

Entropy budget (mW m^{−2} K^{−1}) of the radiative–convective equilibrium model. Dashed arrows represent entropy fluxes of solar radiation, and solid arrows represent fluxes of terrestrial radiation. Numbers in the atmosphere and below the surface represent the net production of entropy due to the solar and terrestrial fluxes. The convective heat flux transports no entropy. The net fluxes in the atmosphere and surface sum to 774 and 505 mW m^{−2} K^{−1}, respectively. Their total of 1279 mW m^{−2} K^{−1} equals the sum of the TOA entropy fluxes to space.

Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0361.1

Entropy budget (mW m^{−2} K^{−1}) of the radiative–convective equilibrium model. Dashed arrows represent entropy fluxes of solar radiation, and solid arrows represent fluxes of terrestrial radiation. Numbers in the atmosphere and below the surface represent the net production of entropy due to the solar and terrestrial fluxes. The convective heat flux transports no entropy. The net fluxes in the atmosphere and surface sum to 774 and 505 mW m^{−2} K^{−1}, respectively. Their total of 1279 mW m^{−2} K^{−1} equals the sum of the TOA entropy fluxes to space.

Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0361.1

Entropy budget (mW m^{−2} K^{−1}) of the radiative–convective equilibrium model. Dashed arrows represent entropy fluxes of solar radiation, and solid arrows represent fluxes of terrestrial radiation. Numbers in the atmosphere and below the surface represent the net production of entropy due to the solar and terrestrial fluxes. The convective heat flux transports no entropy. The net fluxes in the atmosphere and surface sum to 774 and 505 mW m^{−2} K^{−1}, respectively. Their total of 1279 mW m^{−2} K^{−1} equals the sum of the TOA entropy fluxes to space.

Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0361.1

^{−2}K

^{−1}while the surface loses entropy at the rate of 306 mW m

^{−2}K

^{−1}for a total production of 30 mW m

^{−2}K

^{−1}. This material entropy production is present in both the material and control-volume representations. Figure 5 illustrates the irreversible material entropy production.

Schematic diagram of the material entropy production (mW m^{−2} K^{−1}) due to heating. Convection and conduction transport energy from the warm surface to the cooler atmosphere. The entropy gain (indicated by the plus sign) by the atmosphere is greater than the entropy loss (indicated by the minus sign) by the surface because of its lower temperature. As a consequence, the production is positive:

Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0361.1

Schematic diagram of the material entropy production (mW m^{−2} K^{−1}) due to heating. Convection and conduction transport energy from the warm surface to the cooler atmosphere. The entropy gain (indicated by the plus sign) by the atmosphere is greater than the entropy loss (indicated by the minus sign) by the surface because of its lower temperature. As a consequence, the production is positive:

Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0361.1

Schematic diagram of the material entropy production (mW m^{−2} K^{−1}) due to heating. Convection and conduction transport energy from the warm surface to the cooler atmosphere. The entropy gain (indicated by the plus sign) by the atmosphere is greater than the entropy loss (indicated by the minus sign) by the surface because of its lower temperature. As a consequence, the production is positive:

Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0361.1

^{−2}K

^{−1}compared to 24 mW m

^{−2}K

^{−1}for the flux of the incident photons scattered. The difference is a net production of 86 mW m

^{−2}K

^{−1}. Figure 6 illustrates the irreversible entropy production by scattering.

Schematic diagram of the radiation entropy production (mW m^{−2} K^{−1}) due to the (left) absorption and (right) scattering of solar radiation. The entropy produced (plus sign) in the matter at temperature *T* by the absorption of the energy flux (black dashed arrow) of radiation is greater than the entropy flux (blue dashed arrow) of radiation lost (minus sign) from the radiation field:

Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0361.1

Schematic diagram of the radiation entropy production (mW m^{−2} K^{−1}) due to the (left) absorption and (right) scattering of solar radiation. The entropy produced (plus sign) in the matter at temperature *T* by the absorption of the energy flux (black dashed arrow) of radiation is greater than the entropy flux (blue dashed arrow) of radiation lost (minus sign) from the radiation field:

Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0361.1

Schematic diagram of the radiation entropy production (mW m^{−2} K^{−1}) due to the (left) absorption and (right) scattering of solar radiation. The entropy produced (plus sign) in the matter at temperature *T* by the absorption of the energy flux (black dashed arrow) of radiation is greater than the entropy flux (blue dashed arrow) of radiation lost (minus sign) from the radiation field:

Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0361.1

^{−2}K

^{−1}, respectively, while the solar beam loses entropy at the rate of 55 mW m

^{−2}K

^{−1}for a net production of 814 mW m

^{−2}K

^{−1}. Figure 6 illustrates the irreversible entropy production.

*T*that emits a radiation flux of energy

*F*and entropy

*J*, the entropy production is

^{−2}and down the temperature gradient. We note that the associated radiation entropy flux makes no contribution, because the absorption of the radiation cancels with the emission. Similar to the upward convective flux from the surface to the atmosphere, there is an irreversible entropy production, because the atmospheric temperature is less than that of the surface. Figure 7 illustrates the irreversible entropy production by the absorption and emission of terrestrial radiation, as expressed by (5.11).

Entropy production by the emission and absorption of terrestrial radiation summarized by (5.10).

Schematic diagram of the radiation entropy production by the absorption and emission of terrestrial radiation (mW m^{−2} K^{−1}). Black arrows represent energy fluxes; blue arrows represent entropy fluxes. Emission of energy decreases the entropy of the emitting matter (minus sign) but increases the entropy of the radiation field (plus sign). Absorption has the opposite effect. The emissions from the surface and the atmosphere to space are irreversible, as is the net exchange of radiation from the surface to the atmosphere. The net entropy emission is defined by the increase in radiation entropy relative to the loss of material entropy

Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0361.1

Schematic diagram of the radiation entropy production by the absorption and emission of terrestrial radiation (mW m^{−2} K^{−1}). Black arrows represent energy fluxes; blue arrows represent entropy fluxes. Emission of energy decreases the entropy of the emitting matter (minus sign) but increases the entropy of the radiation field (plus sign). Absorption has the opposite effect. The emissions from the surface and the atmosphere to space are irreversible, as is the net exchange of radiation from the surface to the atmosphere. The net entropy emission is defined by the increase in radiation entropy relative to the loss of material entropy

Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0361.1

Schematic diagram of the radiation entropy production by the absorption and emission of terrestrial radiation (mW m^{−2} K^{−1}). Black arrows represent energy fluxes; blue arrows represent entropy fluxes. Emission of energy decreases the entropy of the emitting matter (minus sign) but increases the entropy of the radiation field (plus sign). Absorption has the opposite effect. The emissions from the surface and the atmosphere to space are irreversible, as is the net exchange of radiation from the surface to the atmosphere. The net entropy emission is defined by the increase in radiation entropy relative to the loss of material entropy

Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0361.1

Consistent with (4.7), the net entropy production ^{−2}, ^{−2}, ^{−2}, and ^{−2}. We also find that these contributions are consistent with the Planck approach:

### c. Efficiency analysis

^{−2}. Then the input temperature is the entropic temperature (4.8a) of the solar radiation

^{−2}. The input temperature is unchanged from CV1, but the output temperature is reduced to 191 K, the entropic temperature of the OLR, because the scattered radiation is eliminated from the output temperature definition in (4.8b). The material entropy efficiency is slightly increased. The third case, CV3, excludes the solar radiation from the control volume. The output temperature is again that of CV2 for the OLR, but the input temperature becomes the weighted average of the temperatures where the solar radiation is absorbed:

*T*

_{in}= 275 K. This reduction in input temperature decreases the Carnot efficiency to about a third from the two previous cases. The material entropy production efficiency

*η*

_{mat}is very similar for these three control-volume cases.

Efficiency analysis of the radiative–convective equilibrium model of the climate system (Fig. 2). The climate system is treated as either a control volume (CV) containing matter and radiation or a material system (MS) exclusively. The numerical cases correspond to different assumptions regarding the energy input

^{−2}. In this situation, the input and output temperatures in (2.8) are very similar, and a small Carnot efficiency results. As required by the heat-engine relation (2.9) with no external working, the efficiency factor for the material entropy production equals the Carnot efficiency of 1.0%. Rigorous application of this approach (MS1) is impractical using accurate, nongray calculations where the photon mean-free paths in some spectral lines are less than 1 m.

^{−2}, and the output temperature is the atmospheric temperature of 253 K.

It is worth emphasizing that none of the output temperatures equal the planetary temperature of 255 K that is based on an energy balance. We note that dropping the entropy factor of 4/3 for cases CV2 and CV3 would raise the output temperature to the planetary temperature. However the following simple argument favors a factor greater than unity. The emission of radiation consists of a loss of material entropy with a concomitant gain in radiation entropy. For example, for a body at temperature *T* that emits a radiation flux of energy *F* and entropy *J*, the entropy production is *T*,

We note that the efficiency approach of Johnson (2000) and Lucarini (2009) would include

## 6. Conclusions

The present analysis successfully applies the theory of irreversible thermodynamics to Earth’s climate system. In the steady state, a heat-engine relation (1.4) can be found treating the system as a single material system (section 2) or as a control-volume system (section 4) that combines the material and radiation subsystems below the top of the atmosphere. Because the climate system does no external work, the net interaction between the system and its surroundings results solely in the production of entropy. It is argued that this material entropy production

The efficiency of that interaction ^{−2} K^{−1}. For the incoming solar flux of ^{−2}, the efficiency of Earth’s material entropy production ^{−2} K^{−1},

The current analysis provides a foundation for further research. Improved estimates of Earth’s radiation energy and entropy budgets should be made, along with those for the material entropy budget of the climate system. The material entropy production consists of positive-definite components that allow for intercomparison. For example, Lucarini et al. (2011, their Table 2) estimate that the vertical/convective (as opposed to horizontal/baroclinic) entropy production contributes about 90% of the total production. Alternatively, the dissipation due to falling precipitation (Pauluis and Dias 2012) is comparable to the estimate of the dissipation of kinetic energy in the general circulation (Lorenz 1967). Quantifying the material entropy production will lead to valuable new insights into the climate system.

## Acknowledgments

I thank Eugene Clothiaux, José Fuentes, Jerry Harrington, Seiji Kato, and Sukyoung Lee for constructive exchanges. Dr. Kato suggested the Planck analysis of the entropics appearing in section 5b.

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