JUNE 2017 B A N N O N A N D L E E 1721 Toward Quantifying the Climate Heat Engine: Solar Absorption and Terrestrial Emission Temperatures and Material Entropy Production PETER R. BANNON AND SUKYOUNG LEE Department of Meteorology and Atmospheric Science, The Pennsylvania State University, University Park, Pennsylvania (Manuscript received 15 August 2016, in final form 22 February 2017) ABSTRACT A heat-engine analysis of a climate system requires the determination of the solar absorption temperature and the terrestrial emission temperature. These temperatures are entropically defined as the ratio of the energy exchanged to the entropy produced. The emission temperature, shown here to be greater than or equal to the effective emission temperature, is relatively well known. In contrast, the absorption temperature re- quires radiative transfer calculations for its determination and is poorly known. The maximum material (i.e., nonradiative) entropy production of a planet’s steady-state climate system is a function of the absorption and emission temperatures. Because a climate system does no work, the material entropy production measures the system’s activity. The sensitivity of this production to changes in the emission and absorption temperatures is quantified. If Earth’s albedo does not change, material entropy production would increase by about 5% per 1-K increase in absorption temperature. If the absorption temperature does not change, entropy production would decrease by about 4% for a 1% decrease in albedo. It is shown that, as a planet’s emission temperature becomes more uniform, its entropy production tends to increase. Conversely, as a planet’s absorption temperature or albedo becomes more uniform, its entropy production tends to decrease. These findings underscore the need to monitor the absorption temperature and albedo both in nature and in climate models. The heat-engine analyses for four planets show that the planetary entropy productions are similar for Earth, Mars, and Titan. The production for Venus is close to the maximum production possible for fixed absorption temperature. 1. Introduction conduction, diffusion, and phase changes, and the loss of material entropy by the emission of terrestrial radiation The climate system of a planet is a heat engine that back out to space. In a steady state, the difference between absorbs solar radiation at a relatively high temperature the loss and the gain equals the material entropy pro- and emits terrestrial radiation to space at a lower tem- duction. Thus, the material entropy production budget is perature. Energy is conserved, and no work is done on the intimately connected to the solar and terrestrial radiative planet’s surroundings. The net result of this interaction is heating processes. the production of entropy that, in a steady state, must be The material entropy production is a fundamental exported to space. The entropy production has both a ra- measure of the activity of the climate system. Entropy is diation and a material component. The scattering of the produced by the transport of heat upward from the surface relatively focused, low-entropy, incoming solar radiation to the atmosphere and meridionally from the equatorial to beam into a more diffuse, reflected field and the creation of the polar regions. Entropy is also produced by the viscous an outgoing field of high-entropy terrestrial radiation dissipation of the winds and currents. Entropy is also comprises the radiation entropics of the planet. The ma- produced by the transport of water vapor from the oceans terial entropics involves the sum of the gain in material to the atmosphere and by nonequilibrium phase changes entropy by the absorption of the solar radiation, the irre- and chemical reactions. Thus, material entropy production versible production of entropy by the processes of thermal measuresthevibrancyoftheclimatesystem. Numerous authors have attempted direct quantification Corresponding author e-mail: Peter R. Bannon, bannon@ems. of the various material entropy production processes. An- psu.edu alyses of nonhydrostatic, radiative–convective equilibrium DOI: 10.1175/JAS-D-16-0240.1 Ó 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses). Unauthenticated | Downloaded 10/01/21 05:37 PM UTC 1722 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 74 models (Pauluis and Held 2002a,b; Pauluis and Dias 2012; (1989, 2000) provides a clear procedure to incorporate Singh and O’Gorman 2016) indicate the importance of the spatially and temporally diverse heat inputs and viscous dissipation of the wind and of hydrometeor drag outputs into a global heat-engine analysis. Lucarini along with production due to moist processes, such as (2009) and Lucarini et al. (2014) review these contri- evaporation, diffusion of water vapor, and the cycle of butions. Differences in the analyses often result from melting and freezing. Pauluis and Held (2002a,b)and differences in the specification of the system of interest. Volk and Pauluis (2010) highlight the reduction of viscous Bannon (2015) articulates the need for clear definitions dissipation between dry and moist convection. These an- of the climate system. His example of a simple climate alyses emphasize the entropy production involved in the model yields six different results for the efficiency de- vertical transport of the absorbed solar energy from the pending on the definition of the system. The model surface to the upper atmosphere, where it is emitted back definition dictates the choice of the temperatures as- to space as terrestrial radiation. There is also entropy signed to the entropy produced by the heating. For ex- production associated with hydrostatic processes involved ample, the temperature associated with the solar heating in the meridional transport of energy poleward. Analyses canvaryfromthetemperatureofthesun(;6000 K) to a of hydrostatic, general circulation models include Goody representative tropospheric value (;270 K). Bannon (2000), Lucarini et al. (2011, 2014), and Laliberte et al. (2015) shows that the later temperature, the solar ab- (2015). The production associated with meridional pro- sorption temperature, is the one required to provide cesses appears secondary in importance compared to that direct information on the material entropy production associated with vertical processes. The total atmospheric in the system. entropy production lies in the range of 30–80 entropy The purpose of this manuscript is to provide the tools 2 2 production units (EPU; 1 EPU 5 1mWm 2 K 1). In necessary to quantify the entropy production of a cli- comparison, analyses of oceanic entropy production are mate system through the examination of the gain and relatively small, lying in the range of 1–2 EPU (Gregg loss in entropy due to the absorption of solar radiation 1984; Shimokawa and Ozawa 2001; Huang 2010; Pascale and the emission of terrestrial radiation. Section 2 re- et al. 2011; Bannon and Najjar 2016, manuscript submitted views the formulation of a climate system as a heat to J. Mar. Res.). engine. We present formal definitions of the solar ab- In addition to the direct quantifications, it is also of sorption temperature and the terrestrial emission tem- interest to examine how material entropy production perature. These temperatures define the Carnot efficiency changes as a climate system evolves. Climate studies in- of the system. Section 3 applies the formulation to an ideal dicate that Earth has been globally warmer and cooler climate system (Zircon) that emits uniformly to space at a than present but that the change in temperature is not single emission temperature. Appendix A shows that the necessarily uniform over the planet. In fact, various cli- lower bound for the emission temperature is the effective mate records indicate that, when Earth was warmer, the temperature associated with uniform blackbody emission Arctic was much warmer than the tropics (e.g., Budyko to space. Then use of this temperature provides an upper and Izrael 1991; Hoffert and Covey 1992). The current bound to the material entropy production of the system. warming also shows the same behavior: the Arctic region The maximum production is expressed as a function of the is warming at least twice as quickly as the global average. effective temperature and the absorption temperature. The warming of the Arctic also implies the melting of the The sensitivity of the production to changes in albedo and sea ice and hence a reduction in the polar albedo. In absorption temperature is presented. contrast, the tropical albedo may or may not decrease Section 4 presents an assessment of the role of regional because it is unclear how the tropical cloud cover will variability of the absorption and emission temperatures change. Climate change is not limited to Earth’s climate. on the material entropy production. The motivation for Theories of planetary evolution suggest that Venus once this assessment comes from the aforementioned non- had a more temperate climate before the onset of a uniformity in warming or cooling (e.g., the Arctic tem- runaway greenhouse led to extreme surface temperatures peratures respond much more strongly to global warming of over 700 K. This manuscript seeks to provide a struc- or cooling) and also in albedo changes. For this exercise, ture to quantify the entropy production of a climate sys- we introduce an ideal climate system (Janus) that has tem and its changes. The goal here is to gain insight, but two equal areas with different shortwave and longwave along the way some definitive answers will be provided properties. Janus is a straightforward extension of Zircon while some open questions will be raised. and is the simplest model that allows for the assessment of Heat-engine and entropy analyses of the climate sys- the regional variability. We show that if Janus has two tem have been presented by a number of authors. Peixoto different emission temperatures but the same total emis- and Oort (1992) review the early literature.