1. Introduction
The Lorenz energy cycle (Lorenz 1955) provides an instructive approach to a quantitative investigation of the energetics of the atmosphere. The uneven spatial distribution of diabatic heating in the atmosphere results in an increase in available potential energy that is converted consequently to kinetic energy maintaining the circulation of the atmosphere against friction. Grounded on this theory, creation of kinetic energy at the expense of available potential energy can be decomposed into a contribution from the meridional overturning circulation, denoted as 
The Hadley circulation is identified with rising of warm and moist air in the equatorial region with the descent of colder air in the subtropics corresponding to a thermally driven direct circulation, with consequent net production of kinetic energy. Many studies (e.g., Mitas and Clement 2005, 2006; Frierson et al. 2007; Hu and Fu 2007; Lu et al. 2007; Previdi and Liepert 2007; Seidel and Randel 2007; Seidel et al. 2008; Johanson and Fu 2009; Stachnik and Schumacher 2011; Davis and Rosenlof 2012; Nguyen et al. 2013; Hu et al. 2013) have sought to analyze how the Hadley system has varied under the recent warming climate. Results from these investigations indicate expansion and intensification of the Hadley circulation over the past several decades. A related question is whether the Hadley system has become more energetic.
At the midlatitudes, the circulation of the atmosphere is dominated by wavelike flows. The Ferrel cells represent statistical residues, which result after zonal averaging of large northward and southward flows associated with the quasi-stationary atmospheric waves. The Ferrel cells are identified with the rising motion of relatively cold air at high latitudes and the sinking of relatively warm air at the lower midlatitudes, thus, defining a thermally indirect circulation with consequent consumption of kinetic energy (Peixoto and Oort 1992).
Grotjahn (2003) pointed out that the Carnot cycle concept can be used to estimate the generation of kinetic energy that results from the thermodynamic changes an air parcel undergoes while completing an atmospheric circuit. He estimated the power of one of the Hadley cells by plotting the thermodynamic properties of air parcels on a temperature–pressure diagram. In this study, we extend his approach, investigating the key thermodynamic properties of the Hadley and Ferrel circulations using assimilated meteorological data from MERRA. The analysis allows us to differentiate individual contributions of the Hadley and Ferrel systems to the 
2. Data and methodology
This investigation is based on meteorological data from the MERRA compilation covering the period January 1979–December 2010. Wind speeds, air temperature, and geopotential heights were obtained on the basis of retrospective analysis of global meteorological data using the Goddard Earth Observing System, version 5.2.0 (GEOS-5) Data Assimilation System (DAS). We use the standard monthly output available for 42 pressure levels with a horizontal resolution of 1.25° latitude × 1.25° longitude (Rienecker et al. 2007). The tropical sea surface temperature and ENSO index is from the Goddard Institute for Space Studies (Hansen et al. 1999, 2010).
3. Thermodynamic properties of the Hadley system
a. Illustration of a direct thermal circulation
Values for January 2009 mass streamfunction (109 kg s−1), computed on the basis of the zonal average of monthly assimilated meteorological data, are presented in Fig. 1a. Positive values are indicated by warm (red) colors with negative values denoted by cold (blue) colors. The thick black contour illustrates the direction of motion (white arrows) corresponding to the direct thermal circulation of the Hadley regime. Maximum heating in January 2009 occurs south of the equator. The air is less dense as a consequence, rising due to buoyancy, cooling in the process. Reaching the top of the convection cell, the air moves northward, cooling as it radiates more energy than it absorbs before sinking eventually in the northern subtropics (Marshall and Plumb 2008). The loop is completed as the air moves back across the equator at the surface.
The average meridional circulation of the atmosphere in January 2009 with an emphasis on the direct Hadley system. (a) Color shading specifies the mass streamfunction values (109 kg s−1). The thick black loop with a constant mass streamfunction value of 70 × 109 kg s−1 is chosen as an example to illustrate the thermally driven direct circulation. White arrows point in the direction of the air motion. The (b) P–V, (c) P–ΔV, and (d) T–S diagrams for an air parcel with the example mass streamfunction value.
Citation: Journal of Climate 27, 7; 10.1175/JCLI-D-13-00538.1
As an example of how work is produced by completing travel around one loop of the cell, we selected the segment defined by the thick black contour in Fig. 1a, corresponding to a constant mass streamfunction value of 70 × 109 kg s−1. The pressure–volume (P–V) diagram for the transit of 1 kg of air around this loop is presented in Fig. 1b. At a given pressure level, while the air parcel is experiencing ascending motion, its specific volume (illustrated by the blue line) is always greater than the specific volume (expressed by the red line) associated with the descending portion of the trajectory. The area inside the loop defines the network performed by the air parcel as it completes travel along the indicated loop. The network obtained by completing one circuit of the loop is given by 
According to the second law of thermodynamics, 
b. Power of the Hadley system
Consider two loops with constant streamfunction values, one inside the other (see Fig. S1 in the supplementary material). Assume that the total mass between loops 1 and 2 is represented by M, while the average time required to complete travel through the region sandwiched between loops 1 and 2 is expressed by 
The power of the Hadley circulation as a function of month in (a) the Northern Hemisphere and (b) the Southern Hemisphere, and (c) the total power of both Hadley cells. (d) The variation of the total power of both Hadley cells by year from January 1979 to December 2010.
Citation: Journal of Climate 27, 7; 10.1175/JCLI-D-13-00538.1
From Fig. 2 alone, it is unclear which of the terms in Eq. (5) dominates the annual cycle of Hadley cell power output. Oort and Rasmusson (1970) investigated the annual cycle of the Hadley circulation based on the value of the mass streamfunction and data for a 5-yr period obtained from a dense network of upper-air stations. The seasonality of the Hadley system was explored further by Dima and Wallace (2003) using NCEP–NCAR data covering the period 1979–2001. Defining the seasonality as the principal component of the first EOF mode of the mass streamfunction, Dima and Wallace concluded that the seasonal variation of the circulation is sinusoidal. The functional similarity between these results and the seasonal cycle in Fig. 2 implies that large absolute values of the mass streamfunction are normally associated with large power output of the Hadley system and vice versa.
The long-term variation of the power contributed by both cells is plotted in Fig. 2d, covering the period January 1979–December 2010. The conspicuous intraseasonal fluctuation in the black line reflects the strong seasonal variation of the Hadley circulation. The red line, computed using a 12-month running average, indicates the existence of an interannual variation combined with a longer-term intensification of the circulation. Linear regression of the annual mean average data indicates an increase of 0.54 TW yr−1 in total power since 1979 as defined by the blue line. The associated R2 for the regression analysis however is 0.31, indicating considerable uncertainty in the magnitude of the inferred trend.
c. Thermodynamic efficiency of the Hadley system
As in Fig. 2, but for the efficiency of the Hadley circulation.
Citation: Journal of Climate 27, 7; 10.1175/JCLI-D-13-00538.1
Although the streamfunction term dominates, the thermodynamic efficiency plays an important role as well. Reanalysis datasets typically indicate higher values for the overturning streamfunction value in the Southern Hemisphere as compared to the Northern Hemisphere (Nguyen et al. 2013). However, the Hadley cell in the Northern Hemisphere has a slightly greater peak power with the associated efficiency of 3.2% as compared with the peak power of the corresponding cell in the Southern Hemisphere with an efficiency of 2.3%, reflecting the importance of thermodynamic efficiency in determining the generation of power.
The long-term trend for the overall efficiency of the entire Hadley circulation is plotted in Fig. 3d for the period January 1979–December 2010. The strong fluctuation in the black line reflects the seasonal variation of the circulation. The red line, computed using a 12-month running average, suggests that, at least on an annually averaged basis, the efficiency has varied little over the 30-yr interval covered by the present analysis. Linear regression of the annual mean average over this period provides a regression slope of −0.0029% yr−1 with R2 = 0.06, indicating no statistically significant trend in thermodynamic efficiency.
Monthly values for the heat absorption rates for the Hadley cells in each hemisphere and for the entire Hadley circulation (an average over the entire record covered in this study) are presented in Figs. 4a–c. The long-term variation of the rate at which the heat is absorbed in driving the entire Hadley circulation is plotted in Fig. 4d for the period January 1979–December 2010. The linear regression of the annual mean average data indicates an upward trend with an increase rate of 26.7 TW yr−1 with R2 = 0.55, as shown by the blue line. The interannual variation of the heat absorption rate is associated with variation in tropical sea surface temperature between 23.6°S and 23.6°N (Fig. 5b): high tropical sea surface temperatures (SSTs) correspond to high heat absorption rate, and vice versa. The correlation between the 12-month running average of the heat absorption rate and the tropical sea surface temperature shown in Fig. 5b exceeds 0.6, confirming their strong connection. The ENSO signal is evident also in the heat absorption rate: specifically the warm events in 1983, 1987, and 1997, in addition to the cold events in 1985, 1996, and 1999 (Fig. 5c). The correlation between the heat absorption rate and the ENSO index shown in Fig. 5c is less, 0.32, presumably reflecting the fact that the ENSO phenomenon is more localized in the Pacific region rather than distributed over the entire domain of tropical latitudes.
As in Fig. 2, but for the heat absorption rate of the Hadley circulation.
Citation: Journal of Climate 27, 7; 10.1175/JCLI-D-13-00538.1
Variations of heat absorption rates of (top–bottom) both Hadley cells, low-latitude temperatures, and the ENSO index: (a) 12-month running average of the total heat absorption rate for both Hadley cells from January 1979 to December 2010, (b) changes in mean tropical temperature (data available at http://data.giss.nasa.gov/gistemp) over the period January 1979–December 2010, and (c) Niño-3.4 index (data available at http://www.esrl.noaa.gov/psd/data/climateindices) over the period January 1979–December 2010.
Citation: Journal of Climate 27, 7; 10.1175/JCLI-D-13-00538.1
4. Thermodynamic properties of the Ferrel system
a. Illustration of an indirect thermal circulation
As with section 3, we begin this section by highlighting one specific zonal mean cell. We choose a specific segment shown in Fig. 6a, defined by the thick black contour, with a constant mass streamfunction value of −20 × 109 kg s−1. The P–V diagram for transit of 1 kg around this loop is presented in Fig. 6b. In contrast to the Hadley circulation, while the air parcel is experiencing ascending motion, its specific volume (illustrated by the blue line) is always smaller than the specific volume (expressed by the red line) associated with the descending portion of the trajectory. The network consumed by the air parcel in completing this loop is estimated at 3.43 kJ. The corresponding P–ΔV diagram is presented in Fig. 6c. The T–S cycle in Fig. 6d is much rounder than that in Fig. 1d, reflecting the stronger temperature contrast at midlatitudes, indicating the high efficiency of the Ferrel system in consumption of kinetic energy (Lorenz 1967).
As in Fig. 1, but for the average meridional overturning of the atmosphere in January 2008 with an emphasis on the indirect Ferrel circulation and with the thick black loop having a streamfunction value of −20 × 109 kg s−1.
Citation: Journal of Climate 27, 7; 10.1175/JCLI-D-13-00538.1
b. The power consumption rate of the Ferrel system
With the same approach used to evaluate the key thermodynamic properties of the Hadley circulation, we calculated the power consumption rate, the COP, and heat absorption rate of the Ferrel cells in each hemisphere as well as their combination (Figs. 7–9). The annual mean power consumption associated with the overall Ferrel system amounts to 275 TW, consistent with the conclusion reached by Oort (1983) that the Ferrel system consumes kinetic energy at a rate larger than the rate at which power is produced by the Hadley system. The overall COP of the entire Ferrel circulation is relatively constant, approximately 12.1 for each month, with a relatively small associated variation with season. If the Ferrel system were allowed to circulate in the opposite direction as a thermal engine, its efficiency would be 1/(1 + COP) = 7.6%, significantly greater than that of the Hadley system. The average rate at which heat is absorbed from the cold area by the entire Ferrel circulation over the past 32 years amounts to approximately 3.3 PW. Heat is released at the warmer area of the Ferrel system at a rate of 3.6 PW (3.3 PW + 275 TW).
As in Fig. 2, but for Ferrel circulation.
Citation: Journal of Climate 27, 7; 10.1175/JCLI-D-13-00538.1
As in Fig. 2, but for the COP of the Ferrel circulation.
Citation: Journal of Climate 27, 7; 10.1175/JCLI-D-13-00538.1
As in Fig. 2, but for the heat absorption rate of the Ferrel circulation.
Citation: Journal of Climate 27, 7; 10.1175/JCLI-D-13-00538.1
5. Summary and discussion
The Hadley, Ferrel, and polar circulations all contribute to the zonal mean kinetic energy budget of the atmosphere as illustrated by Fig. 10. The present study indicates an upward trend of the power generated by the Hadley circulation over the past 32-yr period. The analysis suggests that despite the apparent increase in the heat absorption rate, the thermodynamic efficiency of the Hadley circulation has remained relatively constant. Additional input of heat resulted, however, in a net increase in work performed and thus an increase in production of kinetic energy. The increase in the heat absorption rate over the period covered in this study amounted to 26.7 TW yr−1, or 0.1 W m2 yr−1 averaged over the equatorial region dominated by the Hadley circulation (30°S–30°N). The positive trend in the heat absorption rate generally follows the positive trend in surface temperatures observed between 23.6°S and 23.6°N (Fig. 5b).
Lorenz energy cycle with decomposition of the kinetic energy source 
Citation: Journal of Climate 27, 7; 10.1175/JCLI-D-13-00538.1
Regarding the energetics of a Hadley cell, we conclude that, in addition to the absolute value of the mass streamfunction, the thermodynamic efficiency is an important factor in determining the power output. The thermodynamic efficiency is influenced by the profiles of temperature and pressure in the atmosphere. Observational analysis has shown that the Hadley circulation has undergone statistically significant poleward expansion in the past few decades (Hu and Fu 2007). As the Hadley circulation expanded, the temperature and pressure profiles adjusted accordingly. The present results fail to indicate any statistically significant trend in the thermodynamic efficiency.
On the intensification of the Hadley circulation, both Mitas and Clement (2005) and Hu et al. (2005) found evidence for the intensification of the Hadley circulation in the NCEP–NCAR reanalysis. Since large absolute values of the mass streamfunction are normally associated with large power output of the Hadley system, the upward trend in the power output of the Hadley system indicated here is in general agreement with the conclusions from previous studies. Mitas and Clement (2006) pointed out that the trend might reflect systematic observational errors. Hu et al. (2011) argued that the increasing trend in the Hadley circulation strength in ERA-40 might be artificial as well. The MERRA data used in this study were processed in three separate streams. The data distribution adopted here used stream 1 for 1 January 1979–31 December 1992, stream 2 for 1 January 1993–31 December 2000, and stream 3 for 1 January 2001–present. Despite differences in the NCEP–NCAR used in the earlier studies and the MERRA data employed here, conclusions in both cases are in agreement with respect to the temporal intensification of the Hadley circulation.
The Ferrel circulation is an indirect meridional overturning circulation in midlatitudes. The rounder shape of T–S cycle in Fig. 6d as compared to Fig. 1d confirms Lorenz’s 1967 expectation that the stronger horizontal temperature contrast at midlatitudes should enhance the power consumption ability of the Ferrel system. The analysis implies that there has been no statistically significant trend in the power consumption rate of the Ferrel circulation over the past 32 yr.
The contribution of the Hadley and Ferrel circulations in combination have been responsible for net consumption of kinetic energy at an annually averaged rate of 77 TW or 0.15 W m−2 over the past 32 years (Fig. 11). The polar meridional cell is too weak to allow its contribution to be calculated following the procedure adopted here for the Hadley and Ferrel systems. The polar circulation is direct; therefore, it is expected to contribute a net source of kinetic energy. Its contribution is unlikely to significantly offset the net sink attributed here to the combination of the Hadley and Ferrel systems.
The power generated by the combination of Hadley and Ferrel circulations as a function of (a) month and (b) year from January 1979 to December 2010.
Citation: Journal of Climate 27, 7; 10.1175/JCLI-D-13-00538.1
Kim and Kim (2013) analyzed the Lorenz energy cycle using the standard daily output of MERRA dataset covering the period 1979–2008. Based on two different formulations, they estimated 
Peixoto and Oort (1983) pointed out that 
Acknowledgments
The work described here was supported by the National Science Foundation. Junling Huang was also supported by the Harvard Graduate Consortium on Energy and Environment. We acknowledge helpful and constructive comments from Brian F. Farrell, Zhiming Kuang, Michael J. Aziz, and Xi Lu. We are also indebted to two anonymous reviewers for their helpful suggestions.
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