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  • View in gallery

    Contour plot of the maximum material entropy production (EPU) as a function of the planetary solar absorption temperature and terrestrial effective emission temperature for Earth’s solar constant S0 = 1361 W m−2. The emission temperature is a decreasing function of the planetary albedo α. The plus sign indicates the approximate situation for Earth’s climate with = 255 K and ~ 275 K for a production ~ 68 EPU.

  • View in gallery

    Difference in the regional emission temperatures of Janus as a function of the difference in tropical emission temperature from the effective temperature , subject to the constraint that the total OLR is constant. The decrease in the extratropical temperature (blue curve) is greater than the increase in the tropical temperature (red curve). The area average of the two emission temperatures (purple curve) decreases, but the global emission temperature (black curve) increases with increasing temperature difference. The large colored plus signs indicate the values from an analysis of the CERES data (Table 1 and appendix B).

  • View in gallery

    Difference in the regional absorption temperatures of Janus as a function of the difference in tropical absorption temperature from the initial global temperature Ta = 275 K subject to the constraint that the areal-average absorption temperature (purple curve) is constant. The decrease in the extratropical temperature (blue curve) equals the increase in the tropical temperature (red curve). The area average of the two absorption temperatures (purple curve) is constant, but the global temperature (black curve) increases with increasing temperature difference.

  • View in gallery

    Entropy production in percentage of the planetary entropy production constant as a function of the effective emission temperature and absorption temperature. The temperatures are normalized by the planetary temperature . The red dashed line corresponds to . For points above this dashed line, the entropy production decreases with increasing emission temperature (decreasing albedo α). The solar constant is and .

  • View in gallery

    Monthly mean CERES (a) outgoing longwave radiation and (b) absorbed shortwave radiation as a function of time (March 2000–February 2016) for the tropics (red) and the extratropics (blue). Mean values and trends over the 16-yr period are indicated. The trends are computed using the Mann–Kendall method. The superimposed straight lines are the slopes corresponding to the trends. The leftmost values of the straight lines are the corresponding mean values minus the trends multiplied by one-half of the period (8 yr).

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

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  • 1 Department of Meteorology and Atmospheric Science, The Pennsylvania State University, University Park, Pennsylvania
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Abstract

A heat-engine analysis of a climate system requires the determination of the solar absorption temperature and the terrestrial emission temperature. These temperatures are entropically defined as the ratio of the energy exchanged to the entropy produced. The emission temperature, shown here to be greater than or equal to the effective emission temperature, is relatively well known. In contrast, the absorption temperature requires radiative transfer calculations for its determination and is poorly known.

The maximum material (i.e., nonradiative) entropy production of a planet’s steady-state climate system is a function of the absorption and emission temperatures. Because a climate system does no work, the material entropy production measures the system’s activity. The sensitivity of this production to changes in the emission and absorption temperatures is quantified. If Earth’s albedo does not change, material entropy production would increase by about 5% per 1-K increase in absorption temperature. If the absorption temperature does not change, entropy production would decrease by about 4% for a 1% decrease in albedo. It is shown that, as a planet’s emission temperature becomes more uniform, its entropy production tends to increase. Conversely, as a planet’s absorption temperature or albedo becomes more uniform, its entropy production tends to decrease. These findings underscore the need to monitor the absorption temperature and albedo both in nature and in climate models.

The heat-engine analyses for four planets show that the planetary entropy productions are similar for Earth, Mars, and Titan. The production for Venus is close to the maximum production possible for fixed absorption temperature.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author e-mail: Peter R. Bannon, bannon@ems.psu.edu

Abstract

A heat-engine analysis of a climate system requires the determination of the solar absorption temperature and the terrestrial emission temperature. These temperatures are entropically defined as the ratio of the energy exchanged to the entropy produced. The emission temperature, shown here to be greater than or equal to the effective emission temperature, is relatively well known. In contrast, the absorption temperature requires radiative transfer calculations for its determination and is poorly known.

The maximum material (i.e., nonradiative) entropy production of a planet’s steady-state climate system is a function of the absorption and emission temperatures. Because a climate system does no work, the material entropy production measures the system’s activity. The sensitivity of this production to changes in the emission and absorption temperatures is quantified. If Earth’s albedo does not change, material entropy production would increase by about 5% per 1-K increase in absorption temperature. If the absorption temperature does not change, entropy production would decrease by about 4% for a 1% decrease in albedo. It is shown that, as a planet’s emission temperature becomes more uniform, its entropy production tends to increase. Conversely, as a planet’s absorption temperature or albedo becomes more uniform, its entropy production tends to decrease. These findings underscore the need to monitor the absorption temperature and albedo both in nature and in climate models.

The heat-engine analyses for four planets show that the planetary entropy productions are similar for Earth, Mars, and Titan. The production for Venus is close to the maximum production possible for fixed absorption temperature.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author e-mail: Peter R. Bannon, bannon@ems.psu.edu
Keywords: Climate models

1. Introduction

The climate system of a planet is a heat engine that absorbs solar radiation at a relatively high temperature and emits terrestrial radiation to space at a lower temperature. Energy is conserved, and no work is done on the planet’s surroundings. The net result of this interaction is the production of entropy that, in a steady state, must be exported to space. The entropy production has both a radiation and a material component. The scattering of the relatively focused, low-entropy, incoming solar radiation beam into a more diffuse, reflected field and the creation of an outgoing field of high-entropy terrestrial radiation comprises the radiation entropics of the planet. The material entropics involves the sum of the gain in material entropy by the absorption of the solar radiation, the irreversible production of entropy by the processes of thermal conduction, diffusion, and phase changes, and the loss of material entropy by the emission of terrestrial radiation back out to space. In a steady state, the difference between the loss and the gain equals the material entropy production. Thus, the material entropy production budget is intimately connected to the solar and terrestrial radiative heating processes.

The material entropy production is a fundamental measure of the activity of the climate system. Entropy is produced by the transport of heat upward from the surface to the atmosphere and meridionally from the equatorial to the polar regions. Entropy is also produced by the viscous dissipation of the winds and currents. Entropy is also produced by the transport of water vapor from the oceans to the atmosphere and by nonequilibrium phase changes and chemical reactions. Thus, material entropy production measures the vibrancy of the climate system.

Numerous authors have attempted direct quantification of the various material entropy production processes. Analyses of nonhydrostatic, radiative–convective equilibrium models (Pauluis and Held 2002a,b; Pauluis and Dias 2012; Singh and O’Gorman 2016) indicate the importance of viscous dissipation of the wind and of hydrometeor drag along with production due to moist processes, such as evaporation, diffusion of water vapor, and the cycle of melting and freezing. Pauluis and Held (2002a,b) and Volk and Pauluis (2010) highlight the reduction of viscous dissipation between dry and moist convection. These analyses emphasize the entropy production involved in the vertical transport of the absorbed solar energy from the surface to the upper atmosphere, where it is emitted back to space as terrestrial radiation. There is also entropy production associated with hydrostatic processes involved in the meridional transport of energy poleward. Analyses of hydrostatic, general circulation models include Goody (2000), Lucarini et al. (2011, 2014), and Laliberte et al. (2015). The production associated with meridional processes appears secondary in importance compared to that associated with vertical processes. The total atmospheric entropy production lies in the range of 30–80 entropy production units (EPU; 1 EPU = 1 mW m−2 K−1). In comparison, analyses of oceanic entropy production are relatively small, lying in the range of 1–2 EPU (Gregg 1984; Shimokawa and Ozawa 2001; Huang 2010; Pascale et al. 2011; Bannon and Najjar 2016, manuscript submitted to J. Mar. Res.).

In addition to the direct quantifications, it is also of interest to examine how material entropy production changes as a climate system evolves. Climate studies indicate that Earth has been globally warmer and cooler than present but that the change in temperature is not necessarily uniform over the planet. In fact, various climate records indicate that, when Earth was warmer, the Arctic was much warmer than the tropics (e.g., Budyko and Izrael 1991; Hoffert and Covey 1992). The current warming also shows the same behavior: the Arctic region is warming at least twice as quickly as the global average. The warming of the Arctic also implies the melting of the sea ice and hence a reduction in the polar albedo. In contrast, the tropical albedo may or may not decrease because it is unclear how the tropical cloud cover will change. Climate change is not limited to Earth’s climate. Theories of planetary evolution suggest that Venus once had a more temperate climate before the onset of a runaway greenhouse led to extreme surface temperatures of over 700 K. This manuscript seeks to provide a structure to quantify the entropy production of a climate system and its changes. The goal here is to gain insight, but along the way some definitive answers will be provided while some open questions will be raised.

Heat-engine and entropy analyses of the climate system have been presented by a number of authors. Peixoto and Oort (1992) review the early literature. Johnson (1989, 2000) provides a clear procedure to incorporate the spatially and temporally diverse heat inputs and outputs into a global heat-engine analysis. Lucarini (2009) and Lucarini et al. (2014) review these contributions. Differences in the analyses often result from differences in the specification of the system of interest. Bannon (2015) articulates the need for clear definitions of the climate system. His example of a simple climate model yields six different results for the efficiency depending on the definition of the system. The model definition dictates the choice of the temperatures assigned to the entropy produced by the heating. For example, the temperature associated with the solar heating can vary from the temperature of the sun (~6000 K) to a representative tropospheric value (~270 K). Bannon (2015) shows that the later temperature, the solar absorption temperature, is the one required to provide direct information on the material entropy production in the system.

The purpose of this manuscript is to provide the tools necessary to quantify the entropy production of a climate system through the examination of the gain and loss in entropy due to the absorption of solar radiation and the emission of terrestrial radiation. Section 2 reviews the formulation of a climate system as a heat engine. We present formal definitions of the solar absorption temperature and the terrestrial emission temperature. These temperatures define the Carnot efficiency of the system. Section 3 applies the formulation to an ideal climate system (Zircon) that emits uniformly to space at a single emission temperature. Appendix A shows that the lower bound for the emission temperature is the effective temperature associated with uniform blackbody emission to space. Then use of this temperature provides an upper bound to the material entropy production of the system. The maximum production is expressed as a function of the effective temperature and the absorption temperature. The sensitivity of the production to changes in albedo and absorption temperature is presented.

Section 4 presents an assessment of the role of regional variability of the absorption and emission temperatures on the material entropy production. The motivation for this assessment comes from the aforementioned nonuniformity in warming or cooling (e.g., the Arctic temperatures respond much more strongly to global warming or cooling) and also in albedo changes. For this exercise, we introduce an ideal climate system (Janus) that has two equal areas with different shortwave and longwave properties. Janus is a straightforward extension of Zircon and is the simplest model that allows for the assessment of the regional variability. We show that if Janus has two different emission temperatures but the same total emission as Zircon, then there will be less entropy production on Janus than Zircon, all else being equal. Then if Janus evolves with its emission temperatures becoming more uniform, its entropy production will increase. We call this process equalization. Section 4 quantifies this behavior and that for the equalization of the absorption temperatures and the regional albedos.

Section 5 applies the Zircon theory to other planets and estimates the radiation temperatures for Earth, Venus, Mars, and Titan and then compares their entropy productions. Unlike the other planets, the production for Venus is at a maximum for fixed absorption temperature. Section 6 summarizes the results and presents a call for climate modelers to include calculations of their model’s absorption and emission temperatures. These temperatures are essential in analyzing the model’s heat engine and the current and future entropy production.

Using trends in CERES data (analyzed in appendix B), we offer some speculations on future trends in Earth’s entropy production. The purpose of these speculations is not to present hard forecasts but rather to demonstrate the utility of the approach afforded by quantifying the climate system’s heat engine and its entropy production.

2. Heat-engine analysis and material entropy production

A heat-engine analysis requires the merging of the energy and entropy budgets of the thermodynamic system of interest. The thermodynamic system is taken to be the material climate system of the planet and the terrestrial radiation contained therein. This situation corresponds to material climate system case 2 (MS2) in Bannon (2015) and provides direct calculation of the material entropy production. The total energy of this system (the sum of the thermal radiation, kinetic, internal, and potential energies) increases because of the absorption of the flux of solar energy and decreases because of the emission of terrestrial energy out of the system. The energy budget for the total energy E per unit area is
e2.1
where the terms on the right-hand side represent the net absorbed solar flux and the net emitted terrestrial flux . All fluxes (W m−2) are area-averaged quantities. The energy fluxes are defined in terms of the local heating rates per unit volume for solar and terrestrial radiation as
e2.2
where V is the volume of the climate system and is the surface area of the planet. Then the total fluxes in watts are obtained by multiplying the fluxes by the area of the planet. The radiative gain in solar radiation is that due to the absorption of the incoming solar radiation. The radiative loss in longwave radiation is that due to the net emission of the outgoing terrestrial radiation.
The budget for the total entropy S per unit area of the planet is
e2.3
where denotes the rate of entropy gain or loss by a heating process. The irreversible entropy production internal to the planet is . All quantities are area-averaged quantities (W m−2 K−1). The entropy fluxes are related to the energy fluxes by the relations
e2.4a
e2.4b
which define the entropic absorption and emission temperatures. The formulation (2.4) of these radiation temperatures follows Johnson (1989). Using these temperatures, the entropy budget (2.3) may be written as
e2.5
The result (2.5) indicates that the rate of storage of entropy in the climate system is due to the imbalance between the diabatic entropy generation processes and the irreversible entropy production processes.
For a climatological steady state, the time-derivative terms drop from the governing equations: there is no storage of energy or entropy in the climate system. In particular, the entropics is irreversible: the irreversible entropy produced in the climate system is exported radiatively out to space. Then the mathematical analysis is straightforward. The time-averaged energy and entropy equations [(2.1) and (2.5), respectively] reduce to
e2.6
e2.7
which combine to yield an expression for the entropy production as the difference between the net entropy loss by longwave emission and the net gain by solar absorption:
e2.8
or
e2.9
where the Carnot efficiency is
e2.10
The climate system does no external work on its surroundings. Any internal work manifests itself as material entropy production. A fraction of the energy absorbed is involved in the production of entropy.
The effect of energy storage may also be incorporated into the analysis. Hansen et al. (2005) suggest the ocean has been storing energy at the rate of about 1 W m−2. Then we modify the analysis to include this storage, assuming a steady rate of energy and entropy storage . Then (2.1) and (2.5) are replaced by
e2.11
e2.12
respectively. A positive (negative) energy storage may be subsumed into the absorption (emission) term to define a net heating gain (loss). An entropy storage may be subsumed into the entropy production term to define a net production. The effects of storage are not addressed further.

3. Zircon: Maximum material entropy production

We apply the formalism of section 2 to an idealized planet, Zircon, whose climate engine is in a steady state. Introducing the solar constant and the planetary albedo , the energy balance (2.6) is written as
e3.1
where is the Stefan–Boltzmann constant. Here, is the effective emission temperature based on the energy balance (3.1). It is the blackbody emission temperature for which the planet would be in radiative equilibrium. It is shown in appendix A that the maximum loss of entropy from a planet is that due to uniform emission at this effective temperature:
e3.2
Thus, the emission temperature is greater than or equal to the effective emission temperature . As a consequence, the maximum material entropy production (2.8) is, for a given absorption temperature ,
e3.3
Using the energy balance statement, we can also write this maximum as
e3.4
Figure 1 plots the dependence of the maximum entropy production as a function of the solar absorption temperature and the effective temperature (or, equivalently, the albedo) for Earth’s solar constant S0 = 1361 W m−2 (Hartmann 2016). Physically realistic values of positive entropy production are found in the lower-right portion of the diagram for absorption temperatures greater than the effective temperature . The production increases with increasing absorption temperature:
e3.5
But the production decreases with increasing effective temperature
e3.6
provided . The energy balance equation [see (3.1)] indicates that the effective temperature is only a function of albedo for fixed solar constant. Differentiating this relation yields . Then the change in production due to a change in albedo is
e3.7
and the production increases with increasing planetary albedo, provided .
Fig. 1.
Fig. 1.

Contour plot of the maximum material entropy production (EPU) as a function of the planetary solar absorption temperature and terrestrial effective emission temperature for Earth’s solar constant S0 = 1361 W m−2. The emission temperature is a decreasing function of the planetary albedo α. The plus sign indicates the approximate situation for Earth’s climate with = 255 K and ~ 275 K for a production ~ 68 EPU.

Citation: Journal of the Atmospheric Sciences 74, 6; 10.1175/JAS-D-16-0240.1

As a specific example of this analysis, we examine conditions for present-day Earth (Hartmann 2016) with an albedo of 29.3%. Then the effective temperature is 255 K and = 940 EPU. Determination of the absorption temperature requires calculation of both and . The study of Goody (2000) provides an example. The data in his Table 5 ( = 230.7 W m−2 and = 842.5 EPU) implies an absorption temperature of 274 K. His analysis of data from Peixoto et al. (1991) given in his Table 3 suggests an absorption temperature of 278 K. We assume an absorption temperature of 275 K. Then the planetary entropy production is 68 EPU, in broad agreement with analyses of climate models (e.g., Lucarini et al. 2011). The cross in Fig. 1 indicates this estimate for Earth. We find the climate efficiency is .

Based on this analysis, we may also speculate on the trends in entropy production. The production increases with increasing absorption temperature and decreasing effective temperature. The values for the sensitivities of the production to changes in absorption and effective temperatures are
e3.8
e3.9
or, with , the sensitivity to a change in albedo is
e3.10
The finite change in entropy production due to a change in the planet’s climate can be estimated by
e3.11
The CERES data (appendix B and Tables 13) indicate a slight increase in the absorbed solar radiation of 1.47 W m−2 century−1 that corresponds to a trend in the albedo of −0.4% century−1. We caution the reader that the solar radiation trend from CERES is likely affected by natural variability. Nevertheless, we use the trend derived from CERES data to demonstrate the utility of using material entropy production to measure a climate’s vibrancy. We assume that, because the planet has been warming, there has been a general increase in the absorption temperature. If the surface and tropospheric temperatures continue to increase while the albedo decreases as a result of a decrease in polar ice, we estimate a change over the next century of
e3.12
where the change in absorption temperature has been taken to be the same as the change in surface temperature. These estimates suggest an increase in the entropy production of about 10%.
Table 1.

Total, tropical, and extratropical solar heating rates for the globe (GL) and for the Northern and Southern Hemispheres (NH and SH, respectively) based on 2000–16 CERES data, with the ENSO signal removed using a Mann–Kendall significance analysis. The global tropics extend from 30°S to 30°N. The Northern (Southern) Hemispheric tropics extend from the equator to 30°N (30°S). The extratropics lie poleward of the tropics.

Table 1.
Table 2.

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

Table 2.
Table 3.

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

Table 3.

The Zircon model assumes that the emission from the planet is uniform at the effective emission temperature. The next section examines the impact of horizontal variations in the emission and absorption temperatures and in the associated energy fluxes.

4. Janus: Nonuniformity in emission temperature, absorption temperature, and albedo

As was motivated in section 1, in order to analyze the effects of horizontal variations in the emission and absorption processes on the global material entropy production, we introduce a conceptual model: Janus. Janus is a hypothetical planet divided into two regions of equal area with distinctly different radiative properties. In the longwave part of the spectrum, one half the planet emits blackbody radiation at a higher temperature than the other half. Similarly, in the shortwave, one half the planet has a different absorption temperature and albedo than the other half. Janus is in a steady state with no storage. We estimate the effect on the material entropy production, as these radiative properties tend to equalize. We demonstrate that equalization of the emission temperature is a sign of increasing entropy production, but equalization of the absorption temperature or albedo is a sign of decreasing entropy production.

The global emission and absorption temperatures are defined in terms of the regional temperatures and heating by
e4.1a
e4.1b
where the superscript plus and minus signs denote the tropical and extratropical regions, respectively. Regional heating and entropy sources are defined as in (2.2), where the volume of integration is now over each region rather than the whole system. Absorption and emission temperatures for the regions follow in analogy with the global temperatures defined using (2.4). The factor of 2 in (4.1) accounts for the equal areas of the two regions being half that of the whole system.

Figure 2 illustrates some of the consequences of these definitions in terms of the entropic temperatures as inverse, heating-weighted averages. For example, the numerical average of the regional emission temperatures is less than the global effective temperature but the emission temperature is greater than the effective temperature (Fig. 2). This latter fact is consistent with the proof in appendix A that the maximum entropy loss occurs for uniform blackbody emission at the emission temperature. The analysis of Fig. 2 shows the CERES values given in Table 3 with the emission temperatures , , and equal to 255.0, 260.1, and 249.5 K, respectively, while (4.1a) implies an emission temperature Te = 255.15 K. We conclude that using the effective emission temperature for the emission temperature in the Janus model is a good approximation for Earth.

Fig. 2.
Fig. 2.

Difference in the regional emission temperatures of Janus as a function of the difference in tropical emission temperature from the effective temperature , subject to the constraint that the total OLR is constant. The decrease in the extratropical temperature (blue curve) is greater than the increase in the tropical temperature (red curve). The area average of the two emission temperatures (purple curve) decreases, but the global emission temperature (black curve) increases with increasing temperature difference. The large colored plus signs indicate the values from an analysis of the CERES data (Table 1 and appendix B).

Citation: Journal of the Atmospheric Sciences 74, 6; 10.1175/JAS-D-16-0240.1

The energy balance (2.6) implies that a change in the solar absorption is balanced by a change in the terrestrial emission
e4.2
where the change/differential in a quantity is denoted with the symbol d. The differential of the entropy budget (2.8) is
e4.3
Changes in the regional energetics sum to yield the global change. For example,
e4.4
e4.5

a. Entropy production changes inferred from changes in emission

The change in entropy production associated with the change in the planet’s longwave emission is
e4.6
Assuming blackbody radiation fluxes, the change in emission (4.4) is
e4.7
and the entropy change (4.6) is
e4.8
If the total terrestrial emission does not change, then , and we have a statement of equalization:
e4.9
A decrease in tropical emission temperature compensates for an increase in extratropical emission temperature. Then the emission contribution to the change in entropy production (4.8)
e4.10
Assuming a lower emission temperature in the extratropics and the term is positive for polar warming , the entropy production would increase. This result proves the assertion that equalization in the emission temperature may be a sign of increasing entropy production.

Analysis of the CERES data (appendix B and Table 3) yields the values for the terms in (4.8). These results imply an increase in entropy production (4.8) at the rate of about 1.24 EPU century−1. Thus, the analysis suggests that equalization is having a small impact on entropy production. This result is consistent with Fig. 2, which shows the CERES global emission is close to being uniform.

b. Entropy production changes inferred from changes in absorption

The change in entropy production caused by changes in the solar absorption is
e4.11
Using the energy balance (4.5), we find
e4.12
If the total solar absorption does not change, , then we have
e4.13
The first group of terms in (4.13) indicates an increase in entropy production with increasing regional absorption temperatures. We note that the absorption entropy tendency (4.12) can also be expressed in terms of the global absorption temperature as
e4.14
Then using the differential of the expression (4.1b) yields an expression for the change in global temperature relative to the regional changes in temperature and heating:
e4.15
This result indicates that the change in global absorption temperature depends on the change in the regional temperatures and on the redistribution of the solar heating. Increasing the regional temperatures will increase the global temperature. Because , increasing the tropical (extratropical) absorption will increase (decrease) the global temperature.

Figure 3 illustrates some consequences of the definition (4.1) on the absorption temperatures. Here, we use the global tropical and extratropical values of Table 1 and define and , respectively. The regional absorption temperatures are then varied linearly from the assumed global value of 275 K of Zircon. Even though the area-average temperature is that of Zircon, the global absorption temperature is greater for Janus. Equalization of the regional temperatures reduces the global temperature and by (4.14) reduces the entropy production.

Fig. 3.
Fig. 3.

Difference in the regional absorption temperatures of Janus as a function of the difference in tropical absorption temperature from the initial global temperature Ta = 275 K subject to the constraint that the areal-average absorption temperature (purple curve) is constant. The decrease in the extratropical temperature (blue curve) equals the increase in the tropical temperature (red curve). The area average of the two absorption temperatures (purple curve) is constant, but the global temperature (black curve) increases with increasing temperature difference.

Citation: Journal of the Atmospheric Sciences 74, 6; 10.1175/JAS-D-16-0240.1

The second group of terms in (4.13) proves the assertion that albedo equalization may be a sign of decreasing entropy production. Assuming a lower absorption temperature in the extratropics, the second term is negative for increasing polar absorption, and the entropy production would decrease. Because an increase in albedo results in a decrease in absorption, this result proves the assertion.

We estimate the possible effect of changes in the distribution of solar heating and in the absorption temperatures on the entropy production. We assume the CERES solar heating rates of Table 1 and assume the absorption temperatures , , and . For this case, the material entropy production is 68 EPU. The effects of changes in the absorption temperatures and heating on the entropy production are summarized in Table 4. Increases in the absorption temperatures increase the production, with a tropical increase having the larger effect. Increases in the solar heating decrease the production, with an extratropical increase having the larger effect. If an extratropical increase were compensated by a tropical decrease, then a small decrease in the production would occur.

Table 4.

Effect of individual changes in regional absorption temperatures and heating rates on the global absorption temperature and entropy production. The base-state heating rates are given in Table 1, with the absorption temperatures taken to be , , and . For these settings, the entropy production is 68 EPU.

Table 4.

Table 5 presents the effect of the entropy production due to one climate change scenario described by a 2°C increase in the regional solar absorption temperatures along with an increase in the extratropical heating rate of 4.00 W m−2 compensated by a tropical decrease of 0.70 W m−2. These heating trends are those currently evident in the global CERES data (Table 1). The emission temperature is assumed to be 255.20 K. The impact of these changes on the global absorption temperature and entropy production is summarized in Table 5 for various choices of the absorption temperatures. In each case, the global absorption temperature increases by slightly less than 2°C, and the entropy production decreases by less than 0.4 EPU. These results indicate that, for the scenario under consideration, the change in the entropy production is relatively insensitive to the choice of absorption temperature.

Table 5.

Effect on the global absorption temperature and entropy production for various temperatures. The global warming scenario assumes an increase of 2°C century−1 in regional temperatures and extratropical and tropical heating rate trends of 4.00 and −0.70 W m−2 century−1, respectively. The extratropical absorption temperature is adjusted so that the global emission temperature is fixed at 255.2 K. Case B is the benchmark case analyzed in Table 4.

Table 5.

Table 6 compares the change in entropy productions of the Northern and Southern Hemispheres with that of the global results given as case B in Table 5. The opposite tendencies of the solar fluxes between the two hemispheres imply large but opposite trends in the entropy production. It is unclear if the change in each production occurs or if there is an entropy exchange between the hemispheres. If the former case holds, then the trends suggest a large increase in entropy production in the Southern Hemisphere with a large decrease in the Northern Hemisphere.

Table 6.

Trends in the total absorption temperature and entropy production for the global (GL), Northern Hemisphere (NH), and Southern Hemisphere (SH) cases. The global warming scenario assumes an increase of 2°C century−1 in regional temperatures. The extratropical and tropical heating rate trends, based on the CERES data (see Table 1), are indicated in the second and third columns. The total and tropical absorption temperatures are 275 and 285 K in each case. The extratropical absorption temperature is adjusted so that the global emission temperature is fixed. The efficiency and entropy production are 7.2% and 68 EPU for each case.

Table 6.

c. Consideration of temporal variation in the heating and of nongray absorption spectra

It is also worth noting the implication of other climatically relevant situations, such as the diurnal cycle in temperature and the variations in the absorption spectrum. The Janus analysis indicates that equalization of the emission temperatures could be a sign of increasing material entropy production. By drawing an analogy between the spatial division of the tropics and extratropics and the temporal division of day and night, we deduce that a decrease in the diurnal cycle in temperature would be a sign of increasing entropy production. Similarly, by relaxing the gray-radiation approximation, we may draw an analogy between the spatial division and the variation in the absorption spectrum. We then deduce that a change toward a more uniform absorption spectrum (e.g., caused by an increase in greenhouse gas loading in the atmosphere) would also be a sign of increasing entropy production.

5. Earth, Venus, Mars, and Titan

We apply our formulation of Zircon (section 3) to four terrestrial planets in the solar system whose basic radiative attributes have been recently summarized (Schubert and Mitchell 2013). Table 7 and Fig. 4 presents the comparison using a planetary temperature , defined as
e5.1
which is the effective temperature of each planet with zero albedo. Then the maximum entropy production (3.3) may be nondimensionalized using as
e5.2
The data for the solar constants, albedos, heating rates, and atmospheric/surface temperatures are taken directly from Figs. 3–6 of Schubert and Mitchell (2013) for Venus, Mars, and Titan. The absorption temperatures are calculated following (2.4). For example, their Fig. 4 for Venus indicates solar absorption rates of 100, 30, and 20 W m−2 at the temperatures of 250, 400, and 755 K, respectively. Then the total heating is 150 W m−2, and the entropy gain of the Venusian atmosphere is 502.2 EPU, implying an absorption temperature of 299 K. Schubert and Mitchell (2013) incorrectly average the temperatures of absorption (rather than their inverse) to obtain an absorption temperature of 345 K. As a consequence, their entropics and Carnot efficiencies differ from those in Table 6. We take the emission temperature to be the effective temperature for each planet. This choice will slightly overestimate the entropy production and efficiency.
Table 7.

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

Table 7.
Fig. 4.
Fig. 4.

Entropy production in percentage of the planetary entropy production constant as a function of the effective emission temperature and absorption temperature. The temperatures are normalized by the planetary temperature . The red dashed line corresponds to . For points above this dashed line, the entropy production decreases with increasing emission temperature (decreasing albedo α). The solar constant is and .

Citation: Journal of the Atmospheric Sciences 74, 6; 10.1175/JAS-D-16-0240.1

Inspection of Table 7 and Fig. 4 indicates that the planets Earth, Mars, and Titan share general attributes of similar albedo and Carnot efficiency. Venus arises as a distinctly different climate system with high albedo and large efficiency. Venus appears in Fig. 4 to lie close to a “ridge” of maximum entropy described mathematically by (3.6). This relation indicates that the entropy production for fixed absorption temperature is a maximum for an emission temperature three-fourths the absorption temperature: . We note that for this temperature relation the efficiency is exactly 25%.

Physically the presence of the ridge results from competition between the change in entropy production due to a change in solar heating and the associated change in the terrestrial emission temperature. From (2.8), a change in production is described by
e5.3
Production increases because of an increase in solar heating, a decrease in emission temperature, and an increase in absorption temperature. In the steady state, an increase in solar heating implies an increase in the terrestrial emission and a concomitant increase in the emission temperature. Then , and taking , the change in production is
e5.4
Changes in production due to solar heating dominate those due to the increase in emission temperature, provided . The coefficient multiplying the first term in (5.4) is negative for planets above the dashed line in Fig. 4 and positive below.

Consider a situation in which a planetary atmosphere evolves while maintaining its level of entropy production. As its albedo increases, its effective temperature would decrease and, by (5.4), its entropy production would tend to increase. To maintain its entropy production, (5.4) indicates that the absorption temperature would need to decrease. Thus, a planet could maintain its entropy production as its albedo increases by reducing its absorption temperature. Graphically the planet would evolve from its initial position in Fig. 4 with a downward, leftward trajectory. At the ridgeline, a further increase in albedo could be accomplished at constant production only if the absorption temperature did not change. Below the ridgeline, an increase in albedo would reduce the emission temperature but require an increase in absorption temperature. Such an increase, however, would be unphysical. If the atmospheric circulation maintains its intensity and its level of entropy production, the lapse rate of the atmosphere is likely to remain constant. Therefore, at least from the viewpoint of a one-dimensional radiative–convective model, if the emission temperature decreases, the absorption temperature must decrease (Earthlike condition). This analysis suggests that the equilibrium states in the lower-right corner of the entropy production diagram are unstable and that Venus’s atmosphere lies at the extremity of the stable equilibrium regime.

6. Conclusions

A climate system is a heat engine governed by the laws of thermodynamics. Quantifying this engine in terms of its efficiency and entropy production requires determining both the energetics and entropics of the system’s exchange of radiation with its surroundings. This exchange requires knowledge of the solar and terrestrial heating rates and the associated entropy exchanges. Together, this information yields the entropically defined absorption and emission temperatures (2.4) that quantify the engine’s efficiency and entropy production. These temperatures provide essential information and may be used to describe the evolution of a climate system. The comparative planetary climatology of section 5 presents an example.

Another application lies in the area of climate change. Using a heat-engine analysis, the question of climate change can be displayed on a Zircon diagram like Fig. 1. In a steady state, the climate’s entropy production will evolve because of the changes in both the absorption and the emission temperatures. The effective emission temperature is directly related to the planetary albedo. The trend of the planetary albedo in climate model simulations (Bender 2011) indicates decreases in albedo of 1.5% century−1 that imply, using (3.10), a 4-EPU (1 EPU = 1 mW m−2 K−1) or 6% decrease in the entropy production, provided there is no change in the terrestrial entropics. Estimates for global warming suggest surface temperatures increases in the range of 2°–5°C for the next century (IPCC 2014). A concomitant increase in the absorption temperature would imply, using (3.8), a 6–16-EPU or 9%–24% increase in the entropy production. This line of speculation shows why it is important to compute both the absorption and emission temperatures in order to assess the evolution of Earth’s climate system.

The effective emission temperature provides a quick estimate of a lower bound to the emission temperature. Standard radiative calculations of the local solar heating rates should be used along with temperature profiles to determine the local entropy gain due to the absorption. Then the absorption temperature may be determined using (2.4a). Goody (2000) provides the only direct estimate of this kind. We encourage scientists running climate models and reanalyses to make available and/or publish their results of similar calculations of the radiation heating and entropy exchanges. At present, only the NASA MERRA climate simulations (Rienecker et al. 2007) provide some of this essential information.

Complementary to the radiation studies, there should be direct analyses of the material entropy production budget of the climate simulation. The entropy production of the climate system is analogous to the gross domestic product (GDP) of a country’s economy and should be analyzed sector by sector. For example, the entropy production by viscous dissipation provides input on the strength of the kinetic energy generation rate of the climate system. Similarly, the entropy production by heat conduction provides input on the vertical and horizontal transport of thermal energy from the regions of solar absorption to the regions of terrestrial emission to space. Last, the entropy production by the diffusion of water vapor, by nonequilibrium phase changes, and by hydrometeor drag provides input on the strength of the hydrologic cycle of the system. Budgets of these entropy productions calculated directly from the climate simulation should be compared with the radiation gains and losses along with the entropy storage. Such analyses will more fully quantify the simulation of a climate system’s heat engine and its evolution. In order for the model’s entropy budget to be accurate, the physical parameterizations of the model physics should be made “second law” compliant. For example (Gassmann and Herzog 2015), the turbulent fluxes of mass, momentum, and enthalpy should lead to positive definite terms in the model’s entropy equation (along with boundary flux terms).

The emission and absorption temperatures of a climate system quantify its Carnot efficiency and material entropy production. This information provides global insight into a planet’s vibrancy and its evolution.

Acknowledgments

We thank Eugene E. Clothiaux and Jerry Y. Harrington for discussions on entropy loss by longwave emission and on the calculus of variations. We thank the three reviewers, Editor Olivier Pauluis, and Technical Editor Richard Brandt for their constructive comments.

APPENDIX A

Maximization of the Material Entropy Loss by Longwave Emission

We complement the analysis of O’Brien (1997) and present a yardstick for the maximum entropy loss of a climate system due to longwave emission subject to the constraint of energy conservation. Formally, we maximize the time-mean loss subject to the constraint that the time-mean longwave radiation energy loss is the outgoing longwave radiation (OLR) defined by the effective emission temperature . Mathematically, we use the calculus of variations and define the integral :
ea.1
The first integral is the entropy loss due to longwave emission to space, while the term multiplied by the Lagrange multiplier λ is the energy constraint. The last integral in (A.1) defines the energy loss. The Lagrange multiplier is a constant with the dimensions of inverse temperature. The Planck irradiance is a function of temperature T: , where is the Stefan–Boltzmann constant. The gray, absorption pathlength varies from the top of the atmosphere to beyond the lower boundary of the climate system . It is a function of position x and time . The angle is the zenith angle. We assume isotropic emission in local thermodynamic equilibrium. The integration is over the parameters of time t, planetary surface area A, solid angle of outward directions, and emission optical pathlength . For the generic function g of these parameters p, we define its time-mean area-mean definite integral as
ea.2
The limits of the integral with respect to time correspond to the period of interest.
Differentiation of I with respect to the Lagrange multiplier, , yields the constraint that the net mean emission is that of a blackbody at the effective temperature :
ea.3
The variation of the integral I implies an extremum for , which yields the condition
ea.4
This condition implies that the temperature should be constant: . Then the energy constraint (A.3) implies that the uniform temperature is the effective emission temperature . This analysis is limited to a nonscattering atmosphere with a blackbody lower boundary. These constraints insure that only photons emitted upward have a finite probability of escaping the atmosphere.

This analysis indicates that the maximum entropy loss for a given OLR is that from an isothermal climate system. Then the emission temperature is a minimum. In reality, the system is nonisothermal and there are internal components of the longwave emission and absorption processes that comprise an exchange of radiation energy between the warmer and cooler parts of the system. This exchange involves the net heating of the cooler parts and the de-heating of the warmer parts. Thus, the exchange increases the entropy of the climate system. This entropy increase is misleadingly assigned as material entropy production when the effective emission temperature is used as the emission temperature.

APPENDIX B

Analysis Technique of CERES Data

The CERES data (EBAF-TOA Ed2.8; Loeb et al. 2009) consist of the monthly mean outgoing longwave radiation , the reflected solar radiation, and the incoming radiation as a function of latitude and longitude with a resolution of 1°. The difference in the incoming and reflected radiation yields the net absorbed solar radiation . There are 360 × 180 data points centered on the 0.5° points. The data analyzed in this study span March 2000–February 2016, consisting of 192 months of data. To eliminate the annual variation, we obtain yearly time series by averaging consecutive 12-month values: March 2000–February 2001, March 2001–February 2002, etc. This averaging produces 16 data points in time. Hartmann and Ceppi (2014) use the same method in analyzing the CERES data.

To apply the Janus model, we group the data spatially into relatively warm and cool pairings. Three pairings suggest themselves: a warm tropical and cool extratropical pairing for the globe and two similar pairings for each hemisphere. The data are averaged areally over the four regions corresponding to the latitude belts 90°–30°S; 30°S–0°; 0°–30°N; and 30°–90°N. These regions are the tropical and extratropical regions of each hemisphere. The heating for the global tropical and extratropical pairings is displayed in Fig. B1 with dashed lines. In the tropics (red dashed lines), there are notable interannual variabilities, including the anomalously large values during 2015–16. This time period corresponds to one of the strongest El Niño events in recent years. Therefore, for the purpose of evaluating decadal time-scale variations, it is preferable to eliminate the El Niño–Southern Oscillation (ENSO) signal.

Fig. B1.
Fig. B1.

Monthly mean CERES (a) outgoing longwave radiation and (b) absorbed shortwave radiation as a function of time (March 2000–February 2016) for the tropics (red) and the extratropics (blue). Mean values and trends over the 16-yr period are indicated. The trends are computed using the Mann–Kendall method. The superimposed straight lines are the slopes corresponding to the trends. The leftmost values of the straight lines are the corresponding mean values minus the trends multiplied by one-half of the period (8 yr).

Citation: Journal of the Atmospheric Sciences 74, 6; 10.1175/JAS-D-16-0240.1

The elimination of the ENSO signal is achieved by linearly removing the signal using the same method as L’Heureux et al. (2013). Specifically, the ENSO signal in a heating is calculated by linearly regressing the heating against the monthly Niño-3.4 index. [We use the index provided by the Climate Prediction Center (http://www.cpc.ncep.noaa.gov/data/indices/), and the Niño-3.4 data are derived from Extended Reconstructed Sea Surface Temperature, version 4 (ERSST.v4) (Huang et al. 2015).] The ENSO-removed heating is then obtained by subtracting the product of the linear regression coefficient and the Niño-3.4 index from the original time series. The resulting ENSO-removed heating is indicated in Fig. B1 with solid lines. In the tropics, there are often large differences between the original radiative energy fluxes and the corresponding ENSO-removed fluxes. In the extratropics, the differences are very small, as the blue dashed and solid lines are almost indistinguishable.

The trends in Figure B1 show that, for both heatings, the extratropical values (blue curves) increase, and the tropical values (red curves) decrease, indicating that equalizations occur during the analysis period. Evaluation using the Mann–Kendall test (Mann 1945; Kendall 1962), which has been adopted for trend analysis of meteorological data (Shea 2014), indicates that the positive extratropical global trends are significant above the 90% confidence level, while the negative trends in the tropics have lower confidence levels (Tables 1, 2). The trends and the 16-yr-mean values are indicated in the figure. The superimposed straight line in Fig. B1 shows the slope obtained from the Mann–Kendall analysis.

The emission temperatures indicated in Table 3 and Fig. 2 are computed using the ENSO-removed assuming blackbody radiation.

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