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Constrained work output of the moist atmospheric heat engine in a warming climate

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Science  30 Jan 2015:
Vol. 347, Issue 6221, pp. 540-543
DOI: 10.1126/science.1257103

Because the rain falls and the wind blows

Global warming is expected to intensify the hydrological cycle, but it might also make the atmosphere less energetic. Laliberté et al. modeled the atmosphere as a classical heat engine in order to evaluate how much energy it contains and how much work it can do (see the Perspective by Pauluis). They then used a global climate model to project how that might change as climate warms. Although the hydrological cycle may increase in intensity, it does so at the expense of its ability to do work, such as powering large-scale atmospheric circulation or fueling more very intense storms.

Science, this issue p. 540; see also p. 475

Abstract

Incoming and outgoing solar radiation couple with heat exchange at Earth’s surface to drive weather patterns that redistribute heat and moisture around the globe, creating an atmospheric heat engine. Here, we investigate the engine’s work output using thermodynamic diagrams computed from reanalyzed observations and from a climate model simulation with anthropogenic forcing. We show that the work output is always less than that of an equivalent Carnot cycle and that it is constrained by the power necessary to maintain the hydrological cycle. In the climate simulation, the hydrological cycle increases more rapidly than the equivalent Carnot cycle. We conclude that the intensification of the hydrological cycle in warmer climates might limit the heat engine’s ability to generate work.

As a reflection of the seminal work of Carnot, atmospheric motions have been described as an important component of the planetary heat engine (1). The concept of a heat engine is closely associated with the idea of work: For two cycles that transport the same amount of heat between the same two reservoirs, the one that generates the least irreversible entropy will produce the most work (2). We quantify the atmosphere’s work output through a budget of its entropy production. Previous attempts at obtaining such a budget either resulted in gross estimates (3, 4) or required highly specific data from climate models for a precise analysis (58). Some of these studies showed that the hydrological cycle was an important contributor to the generation of irreversible entropy (6, 9, 10), suggesting that moist processes, including the frictional dissipation associated with falling hydrometeors (11, 12), tend to limit the work output of the atmospheric heat engine. On a warming Earth, the increase in precipitable water (13) has been identified as a reason for the tropical overturning to slow down (14), and studies over a wide range of climates suggest that global atmospheric motions are reduced in extremely warm climates (1517). Models forced according to a climate change scenario also exhibit this behavior in their tropical circulation (18). Here, we employ a method that uses high-frequency and high-resolution data to obtain an atmospheric entropy budget from climate models and reanalyses. This method does not depend on specialized model output, making the diagnostic applicable to the suite of models produced for the Climate Model Intercomparison Project phase 5 (CMIP5) and paving the way to a systematic analysis of the entropy budget in climate models, as proposed by some authors (8).

We base our analysis on the first law of thermodynamics describing moist air (19, 20).Embedded ImageThe material derivatives of moist enthalpy h and moist entropy s (1921) are given by Embedded Image and Embedded Image, respectively.

The equation of state used here provides a comprehensive treatment of moist thermodynamics, including the effect of the latent heat of fusion on Embedded Image and Embedded Image (19, 20). The specific ratio Embedded Image represents the total mass of water divided by the total mass of wet air (humid air plus water condensate). The work output Embedded Image is given by the product of the specific volume Embedded Image with the vertical velocity in pressure coordinates Embedded Image. The chemical potential μquantifies the effect of adding or removing moisture; it is equal to the sum of two terms with different physical meanings (see the supplementary text). The first of these terms accounts for the moistening inefficiencies that accompany the irreversible entropy production associated with the addition of water vapor to unsaturated air (9, 10). The second term accounts for the enthalpy changes associated with the drying and moistening of air. For the atmospheric thermodynamic cycle, we will show that when Embedded Image is positive, Embedded Image primarily quantifies the moistening inefficiencies accounted for by the first term, and when Embedded Image is negative Embedded Image primarily quantifies how much power is associated with combined moisture and dry air fluxes between the surface and the precipitation level accounted for by the second term.

Averaging the first law using a mass-weighted annual and global spatial mean [denoted as {□}] results in simplification. First, Embedded Image equals the difference between interior moist enthalpy sinks and the moist enthalpy sources at Earth’s surface stemming from diffusive fluxes. If we assume that the atmospheric system is in steady state (and therefore approximately yearly periodic), the sinks cancel the sources and Embedded Image vanishes. Moreover, under this averaging, Embedded Image is positive because it quantifies the power necessary to maintain the hydrological cycle and accounts for the moistening inefficiencies (10), and Embedded Image is also positive because it is associated with the dissipation of kinetic energy at the viscous scale (22). Writing Embedded Image, then, the first law reads [equation 4 in (10)]Embedded ImageEmbedded Image is thus reduced by the moistening inefficiencies accounted for by Embedded Image. In the following sections, we obtain a diagnostic for Embedded Image and Embedded Image based on the area occupied by the atmospheric thermodynamic cycle in a temperature-entropy diagram (hereafter Embedded Image diagram) and in a specific humidity-chemical potential diagram (hereafter Embedded Image diagram), respectively.

We analyze two different data sources. The first source is a coupled climate model simulation using the Community Earth System Model (CESM) version 1.0.2 (23). The time period 1981 to 2098 is simulated using a combination of historical radiative forcing estimates and the Representative Concentration Pathway 4.5 (24) future scenario (hereafter historical RCP45). The second source is the period 1981 to 2012 of the Modern-Era Retrospective Analysis for Research and Applications (MERRA) reanalysis (25). For these two data sets, we use a recently developed method (20) to project the material derivative Embedded Image from eulerian space to Embedded Image, its representation in the Embedded Image diagram. We use the same method to project Embedded Image to Embedded Image, its representation in the Embedded Image diagram. Each of these quantities is associated with a closed, uniquely defined mass flux stream function in its respective coordinate system.Embedded Image Embedded ImageEach stream function describes a separate aspect of the large-scale atmospheric thermodynamic cycle. This approach is similar to a method that has been previously used to study the atmospheric and oceanic circulations in thermodynamic coordinates (2629).

In a Embedded Image diagram, the quantity Embedded Image describes a clockwise cycle (Fig. 1, A and B) with three main branches. In the lower branch, a large fraction of air is transported along the surface saturation curve (1000 hPa, 100% relative humidity) and, as it moves toward warmer temperatures, picks up heat through exchanges at Earth’s surface. In the tropical branch, air is transported from warm temperatures to colder temperatures at almost constant moist entropy along the zonal-mean tropical (15°S to 15°N) profile. The zonal-mean tropical profile thus represents the transformations that tropical air masses undergo when they convect, detrain, and mix with environmental midtropospheric air masses (30). In the third branch, radiative cooling acts to reduce entropy as air is transported from high moist entropy and cold temperatures to low moist entropy and warmer temperatures. The thermodynamic cycles in CESM and MERRA have a similar shape, but MERRA’s is stronger (larger maximum stream function value). In the region of the Embedded Image diagram rightward of the zonal-mean subtropical profile (25°N to 35°N) and at temperatures lower than 280 K, the saturation specific humidity is small, so that pressure is approximately a function of temperature and moist entropy. In this region, thermodynamic transformations that occur along discrete pressure levels [diagnosed by Embedded Image, Embedded Image, and Embedded Image (20) but not explicitly demonstrated here] yield the prominent sawtooth patterns seen in the CESM cycle. Although the same patterns appear in the MERRA cycle, they are not as prominent because the vertical resolution is finer by a factor of three, so that a smaller portion of the thermodynamic cycle is sampled along a given pressure level.

Fig. 1 Thermodynamic diagrams for years 1981–2012.

(A and C) CESM. (B and D) MERRA. [(A) and (B)] Embedded Imagediagram. Embedded Image in color shading and gray contours (–75, –225, –375, and –525 Sv). Thick dashed lines represents the zonal-mean vertical profile in different latitude bands. Dotted lines give the 100% relative humidity curves at 1000 hPa, 500 hPa, and 200 hPa. Axes are oriented so that the lower left corner is closest to typical tropical surface Embedded Image values, the upper left corner is closest to typical tropical upper tropospheric Embedded Image values, and the right side of the graph is closest to typical polar Embedded Image values. The correspondence between the equivalent potential temperature Embedded Image and Embedded Image (21) is indicated on the top abscissa. [(C) and (D)] Embedded Image diagram. Embedded Image in color shading and gray contours (–333, –667, –1000, and –1333 Sv). The thick dashed line represents the zonal-mean vertical profile in the tropics. Pressure levels are indicated along this profile. Axes are oriented so that the lower left corner is closest to typical tropical surface Embedded Image values. [(A) to (D)] Thin black curves indicate where Embedded Image= –1 Sv or Embedded Image= –1 Sv. They indicate the small-magnitude cutoff over which Embedded Image or Embedded Image were computed to avoid floating-point errors.

Another difference between the CESM and MERRA thermodynamic cycles is found along the zonal-mean tropical profile. At low temperatures, MERRA’s Embedded Image is weaker than that of CESM, which suggests that MERRA’s thermodynamic cycle represents a tropical heat engine more confined to the lower half of the troposphere.

The power associated with the motions described in the T − s diagram can be computed as the area occupied by the stream function Embedded Imagein a similar way that one would quantify the work output of an equivalent Carnot cycle.Embedded ImageThe averages of Embedded Image over the 1981 to 2012 period for MERRA (6.34 Wm−2) and for CESM (6.36 Wm−2) have a similar magnitude (Fig. 2, A and C).

Fig. 2 Time series of 1-year running mean.

(A) Embedded Image CESM. (B) Embedded Image CESM. (C) Embedded Image MERRA. (D) Embedded Image MERRA. Gray shadings quantify Embedded Image.

In a Embedded Image diagram, the quantity Embedded Image describes a clockwise cycle (Fig. 1, C and D) restricted to Embedded Image≥ 0 comprising three main branches. In the dry branch, air is transported from μ = 0 to low relative humidity (high μ) near the Embedded Image= 0 line. In the moistening branch, air is transported from low Embedded Image and low relative humidity to high Embedded Image at saturation. The cycle to the right of the μ= 0 line, formed by the dry and moistening branches, primarily quantifies the moistening inefficiencies (as can be seen by comparing Fig. 1, C and D, with fig. S1). At high Embedded Image the moistening branch follows the zonal-mean tropical (15°S to 15°N) profile from 600 hPa down in the CESM but only from 800 hPa down in MERRA. Because of this important difference, moistening air masses between 800 hPa and 600 hPa are drier than the tropical environment in MERRA but not in CESM, leading to fewer moistening inefficiencies in CESM. This difference in how air masses are moistened could arise if, for example, the data assimilation in MERRA corrected the tendency of parameterized convection to detrain and mix with its environment. In both cases, the tropical environment nevertheless determines the highest Embedded Image values that can be produced by the atmospheric circulation.

In the dehumidification branch, air is kept close to μ ≈ 0 and transported from high Embedded Image to low Embedded Image. We note that in MERRA, the dehumidification branch reaches into negative μ values between Embedded Image = 5 g kg−1 and Embedded Image = 10 g kg−1, whereas the dehumidification branch in CESM follows the μ = 0 line (thin dashed line) more closely in this region of the Embedded Image diagram.

The power necessary to maintain the global moistening and remoistening motions described in this diagram can also be written as the area occupied by the stream function Embedded Image.Embedded ImageThe average Embedded Image over the 1981 to 2012 period for MERRA (2.68 Wm−2) (Fig. 2D) is 18% larger than for CESM (2.27 Wm−2) (Fig. 2B). The difference in Embedded Image is mostly explained by a reduced amplitude of moistening inefficiencies in CESM (1.97 Wm−2) (fig. S2B) as compared to MERRA (2.40 Wm−2) (fig. S2D). Because the values of Embedded Image are similar for the MERRA and the CESM data, but those for Embedded Image differ, values for Embedded Image (Fig. 2, A and C) differ when averaged over the period (3.66 Wm−2 for MERRA, 4.09 Wm−2 for CESM). Computing Embedded Image from an Embedded Image diagram gives the same results and shows that the CESM has a larger work output between 500 hPa and 200 hPa (fig. S3). This suggests that the smaller amplitude of moistening inefficiencies in CESM might allow more deep convection than in MERRA.

The climate change response of Embedded Image (Fig. 3A) to an anthropogenic forcing scenario takes the shape of a dipole centered along the zonal-mean tropical profile. The mass flux stream function strengthens at higher moist entropy values and weakens at lower moist entropy values, indicative of a translation to higher moist entropy values. The same response is observed for the zonal-mean profiles.

Fig. 3 Response of CESM thermodynamic diagrams between years 1981 to 2012 and years 2067 to 2098.

(A) Response of Embedded Image (color shading). Gray contours indicate the 1981 to 2012 streamlines (Fig. 1A). Thick dashed lines show the change in zonal-mean vertical profiles for different latitude bands between the 1981 to 2012 (black) and the 2067 to 2098 (blue) periods. Dotted lines illustrate the 100% relative humidity curves at 1000 hPa, 500 hPa, and 200 hPa. (B) Response of Embedded Image (color shading). Thick dashed lines (black and blue overlapping) as in (A) but only for the tropical (15°S to 15°N) profile. Gray contours indicate the 1981 to 2012 streamlines (Fig. 1C).

The response of Embedded Image (Fig. 3B) does not exhibit a dipole as in the Embedded Image diagram. Instead, it is strengthened almost everywhere in the Embedded Image diagram by a mostly uniform value. Accordingly, the zonal-mean tropical profile does not move appreciably. The strengthening is particularly concentrated along the outermost streamline, suggesting a slight dilatation of the cycle that results in more moistening happening at higher Embedded Image toward the end of the 21st century.

By changing the footprint and magnitude of Embedded Image and Embedded Image, the effect of anthropogenic forcing will also influence Embedded Image. Both Embedded Image and Embedded Image (Fig. 4A) exhibit large interannual variability of comparable magnitude but substantially smaller interdecadal variability. Both sets of time series demonstrate that, while both Embedded Image and Embedded Image increase in response to projected climate change, Embedded Image increases more rapidly.

Fig. 4 Evolution of the different components of the first law over the 21st century.

(A) Difference of Embedded Image (red, 1-year running mean in pale, 10-year running mean in solid) and Embedded Image (blue, 1-year running mean in pale, 10-year running mean in solid) from the 1981 to 2012 control period. Theoretical scaling of Embedded Image according to the increase in global specific humidity on the model level nearest to the surface in green (1-year running mean in pale, 10-year running mean in solid). (B) Embedded Image (1-year running mean in gray, 10-year running mean in black), its trend (black line), and the 95% one-sided t test CI for the trend (gray shading).

The evolution of Embedded Image in response to anthropogenic forcing indicates a trend of –0.038 ± 0.08 Wm−2 [one-sided t test, 95% confidence interval (CI)] per 100 years for the 10-year running mean (Fig. 4B). Over the 21st century, this amounts to a small reduction in Embedded Image (≈–1%). In an Embedded Image diagram, the Embedded Image response at the end of the 21st century exhibits a reduction of –0.102 Wm−2 per 100 years below 300 hPa and an increase of 0.073 Wm−2 per 100 years above 300 hPa (fig. S4); at 500 hPa, it is reduced by more than 5% per 100 years. This response is compatible with a general weakening of tropospheric motions accompanied with a strengthening of deep convective motions that have enough energy to reach the upper troposphere.

Because Embedded Image is proportional to the material derivative Embedded Image, we might expect its response to scale like the near-surface specific humidity and therefore be directly related to changes in global surface temperature through a surface Clausius-Clapeyron scaling. Indeed, the specific humidity on the model level nearest to the surface explains 94% of the variance of Embedded Image over the entire 1981 to 2099 period (Fig. 4A). Both the near-surface specific humidity and Embedded Image increase by 5.4% per K global surface warming, which is slightly less than the 7% per K surface warming increase associated with a tropical Clausius-Clapeyron scaling (14). In fact, most of the increase in Embedded Image can be attributed to an increase in the moistening inefficiencies (fig. S5A). Moreover, this increase alone compensates for the increase in Embedded Image (fig. S5B) and is therefore sufficient to constrain the atmospheric heat engine’s work output.

Previous theoretical analyses of the entropy cycle (6, 31) suggest that Embedded Image should scale like Embedded Image, where Embedded Image is the outgoing long-wave radiation and Embedded Image, Embedded Image, and Embedded Image are the mean temperature of atmospheric heat input, output, and dissipation, respectively. Observations of recent tropospheric warming [figures 2.26 and 2.27 in (32)] show that temperature trends are somewhat uniform in the vertical, which suggests that the difference Embedded Image might increase more slowly than either Embedded Image or Embedded Image. This slower increase may explain why Embedded Image does not follow a surface Clausius-Clapeyron scaling and why one would expect moist processes to limit the work output in simulations with anthropogenic forcing. Simulations over a wider range of climates would help verify this hypothesis.

Our comparison of thermodynamic cycles in CESM and MERRA show many similarities; however, we find that CESM requires less power to maintain its hydrological cycle than the reanalysis, due to the smaller amplitude of its moistening inefficiencies. We suggest that this difference might be a consequence of the idealized nature of parameterized convection schemes, and it is likely that it might also influence the response of CESM to anthropogenic forcing. Typically, convection schemes artificially transport moisture along a moist adiabat without accounting for the work needed to lift this moisture, but in the real world, this work is necessary to sustain precipitation. Any increase in global precipitation therefore requires an increase in work output; otherwise, precipitation would have to become more efficient, for example, by reducing the frictional dissipation of falling hydrometeors (11, 12). This is one reason we should interpret the constraint in work output in CESM as a constraint on the large-scale motions and not on the unresolved subgrid-scale convective events.

Our work illustrates a major constraint on the large-scale global atmospheric engine: As the climate warms, the system may be unable to increase its total entropy production enough to offset the moistening inefficiencies associated with phase transitions. This suggests that in a future climate, the global atmospheric circulation might comprise highly energetic storms due to explosive latent heat release, but in such a case, the constraint on work output identified here will result in fewer numbers of such events. Earth’s atmospheric circulation thus suffers from the “water in gas problem” observed in simulations of tropical convection (6), where its ability to produce work is constrained by the need to convert liquid water into water vapor and back again to tap its fuel.

Supplementary Materials

www.sciencemag.org/content/347/6221/540/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S5

References (3336)

References and Notes

  1. Materials and methods are available as supplementary material on Science Online.
  2. Acknowledgments: We acknowledge the Global Modeling and Assimilation Office (GMAO) and the Goddard Earth Sciences Data and Information Services Center (GES DISC) for the dissemination of MERRA data. This work was supported by the G8 Research Initiative grant “ExArch: Climate analytics on distributed exascale data archives” made available through the Natural Sciences and Engineering Research Council (NSERC).
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