1. Introduction
Intense winds in hurricanes and typhoons require a continuous generation of kinetic energy within the storm to balance frictional dissipation. The hurricane circulation transports energy received from the warm ocean to the colder atmosphere. In doing so, it acts as a heat engine that produces the kinetic energy necessary to sustain the storm. The ability to generate kinetic energy can be quantified by an efficiency defined as the fraction of the heat input that is converted into kinetic energy. The efficiency depends on multiple environmental factors, such as the temperature of the energy source and sink, or the relative humidity of the air. In this paper, we will review these factors and show how to assess the efficiency for storms simulated in high-resolution atmospheric models.
The Carnot cycle is probably the best known theoretical model for a heat engine. Its efficiency is the maximum efficiency of any closed thermodynamic cycle and is equal to the ratio of the temperature difference between the heat source and sink to the absolute temperature of the heat source. Hurricanes have at times been compared to a Carnot cycle (Emanuel 1986, 2003; Willoughby 1999) in which the energy source is the warm ocean surface and the energy sink corresponds to the radiative cooling of the troposphere. For a typical ocean temperature of about 300 K and tropopause temperature of 200 K, hurricanes would be able to convert up to one-third of the energy input into kinetic energy.
However, not all heat engines act as Carnot cycles. There is a growing body of evidence that the hydrological cycle leads to a substantial reduction of the generation of kinetic energy by Earth’s atmosphere. This occurs for two reasons. First, a substantial fraction of the work done by the atmosphere is used to lift water and is subsequently dissipated as precipitation falls to the ground (Pauluis et al. 2000; Pauluis and Dias 2012). Second, the atmosphere acts as a dehumidifier that gains water through evaporation in unsaturated air but loses it as liquid water. This corresponds to a thermodynamic transformation in which the reactant (water vapor) has a lower Gibbs free energy state than the product (liquid water or ice). Such reaction cannot occur spontaneously in an isolated system and reduces the ability of the system to generate mechanical work (Pauluis 2011). Several studies (Pauluis and Held 2002a; Laliberté et al. 2015; Pauluis 2016) have confirmed the negative impacts of the hydrological cycle on the atmospheric heat engine efficiency at both the convective and global scales.
This raises the questions of whether hurricanes can generate kinetic energy at a rate expected from a Carnot cycle, and, if so, of why hurricanes would be less affected by moist processes than other atmospheric motions. To address this issue, we will analyze the thermodynamic behavior of an idealized hurricane simulation. Computing the mechanical output of a thermodynamic cycle is straightforward for idealized cycles. This task is more difficult for highly turbulent flows in which the trajectories of air parcels vary greatly and are not periodic. To address this problem here, we use a new analytical framework, the mean airflow as Lagrangian dynamics approximation (MAFALDA; see Pauluis 2016). Under MAFALDA, one first computes the overturning circulation in isentropic coordinates by sorting rising and descending air parcels in terms of their equivalent potential temperature. This mean circulation is then used to construct a set of thermodynamic cycles with the same mass and heat transport as the total flow. The thermodynamic transformations along these cycles are then analyzed to assess the impacts of moist processes on kinetic energy generation in the hurricane.
Section 2 reviews the impacts of the hydrological cycle on the kinetic energy generation in a generic thermodynamic cycle with condensation and precipitation. It shows that the mechanical output of such a cycle is reduced by a Gibbs penalty term that accounts for the addition and removal of water substance in different thermodynamic states. Section 3 describes the MAFALDA procedure and applies it to a hurricane simulation. Section 4 analyzes the thermodynamic cycles in our simulation to show that the thermodynamic cycle associated with ascent within eyewall can achieve an efficiency comparable to that of a Carnot cycle. Our results are summarized in section 5.
2. Impacts of the hydrological cycle on the atmospheric heat engine
We consider a schematic representation of the overturning circulation in a hurricane as presented in Fig. 1. As air rushes toward the center of the storm (points 1 → 2), it gains energy and entropy owing to the energy flux from the surface. It then ascends in the eyewall, undergoing a near-adiabatic expansion, and moves away from the storm center in the upper troposphere (points 2 → 3). The air is eventually brought back to the surface while losing energy through the emission of infrared radiation (points 3 → 1). These transformations correspond to a heat engine that transports energy from the ocean surface to the upper troposphere and is associated with a net conversion of internal energy into kinetic energy.
Schematic representation of a hurricane as a heat engine. Steps 1 → 2: the low-level inflow gains energy from the ocean surface 
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-16-0353.1
In the idealized cycle, water vapor is added as unsaturated water vapor and removed mostly as liquid water or ice. The Gibbs free energy of unsaturated water vapor is always less than that of liquid water at the same temperature with 
A physical process, such as condensation of unsaturated water vapor, in which the Gibbs free energy of the products is higher than that of the reactants, cannot occur under isothermal and isobaric conditions, as it would imply a violation of the second law of thermodynamics. Indeed, in such a situation, the reverse reaction (e.g., the evaporation of liquid water in unsaturated air) occurs spontaneously. As the transformations involved in the idealized hurricane cycle described in Fig. 1 are neither isothermal nor isobaric, they can result in a net increase in the Gibbs free energy without violating the second law. However, Eq. (10) indicates that, when this happens, the cycle must be associated with a heat transport from warm to cold, and the mechanical output is reduced by an amount equal to the Gibbs penalty.
The difference of Gibbs free energy between water vapor and liquid water, 
3. Reconstruction of thermodynamic cycles from numerical simulations
a. Numerical model and setup
We analyze here the thermodynamic behavior of a hurricane simulated with the Advanced Research version of the Weather Research and Forecasting (WRF-ARW) Model, version 3.1.1 (Skamarock et al. 2008). In this configuration, the model uses three two-way nested domains, with respective sizes of 4320 km by 4320 km, 1440 km by 1440 km, and 720 km by 720 km, and with horizontal grid spacings of 18, 6, and 2 km. The model has 41 vertical levels with the model top at 50 hPa. The two smaller nested domains are moveable, with the domain center following the 850-hPa center of the tropical cyclones. The physical parameterizations are as in Zhang and Tao (2013) and Tao and Zhang (2014). It should be noted that the turbulent parameterization used in WRF does not include a frictional heating—that is, the kinetic energy loss to dissipation is not put back as internal energy. Bister and Emanuel (1998) have suggested that the inclusion of frictional heating can lead to more intense tropical storms. The model is initialized with a modified Rankine vortex with a maximum surface wind speed of 15 m s−1 at 135-km radius. The Dunion non–Saharan air layer mean hurricane season sounding (Dunion 2011) is used for the environmental moisture and temperature profile with a constant sea surface temperature of 29°C (SST29) and a constant Coriolis parameter equivalent to 20°N. The initial condition and model setup are as in the noflow-SST29 experiment in Tao and Zhang (2014) but without moisture perturbation.
Figure 2 shows the evolution of the maximum wind and minimum pressure. The hurricane reaches its maximum intensity by the end of day 5, with a central pressure of 885 hPa and a maximum wind speed of 97 m s−1. The storm maintains its intensity for the remaining 10 days of simulation, with a slight increase in surface pressure by day 15. As the experimental setup used here does not include radiative transfer, the atmosphere will slowly evolve toward a state of a thermal equilibrium with the ocean, with no convection or wind. Over the course of the simulation, we observe an increase in low-level humidity away from the storm, a warming of the upper troposphere, and a reduction of the convective activity far away from the storm center. All these are consistent with a slow evolution toward thermal equilibrium. The storm however occupies only a small fraction of the domain and, as noted earlier, its intensity remains steady for the last 10 days of the simulation. Our main focus here is to analyze the thermodynamic cycles that underlie the storm, and we chose here to focus primarily on the intensifying storm on day 5 of the simulation.
(top) Minimum pressure and (bottom) maximum tangential wind.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-16-0353.1
Figure 3a shows the mean azimuthal wind during the fifth day of the simulation. It exhibits a well-defined maximum near the surface at a radius of about 40 km from the storm center. The strong vortex extends through the entire troposphere. Farther away from the center, in the upper troposphere, the circulation is anticyclonic, as evidenced by the negative azimuthal wind.
Time and azimuthal average of the (a) tangential wind, (b) equivalent potential temperature, and (c) streamfunction. The solid black line and the dashed blue line correspond to trajectories associated with the inner-core cycle and rainband cycle, respectively (see section 4). In (a), three locations have been marked along each trajectory: point 1 is the lowest entropy value, point 2 indicates the highest entropy near the surface, and point 3 is the highest point along the cycle.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-16-0353.1
Figure 3b shows the distribution of equivalent potential temperature 
b. The mean airflow as Lagrangian dynamics approximation
The analysis of the thermodynamic cycles in the previous section requires us to know the evolution of the thermodynamic properties of an air parcel. Most atmospheric flows are highly turbulent, and not only are all parcel trajectories different, but they almost never correspond to a closed thermodynamic cycle. To circumvent this problem, Pauluis (2016) introduced MAFALDA, a systematic approach designed to extract a set of representative cycles from numerical simulations of turbulent atmospheric flows. The method consists of four distinct steps:
compute the isentropic streamfunction in z–θe coordinates,
estimate the conditional average of thermodynamic state variables as function of z and θe,
construct a set of trajectories in z–θe coordinates from the isentropic streamfunction, and
interpolate the values of the various state variables along these trajectories.
1) Isentropic streamfunction
Isentropic streamfunction in z–θe coordinates. The inner-core cycle (solid black line) and the outer cycle (dashed blue line) correspond to two isolines of the streamfunction. Three locations have been marked along each trajectory: point 1 is the lowest entropy value, point 2 indicates the highest entropy near the surface, and point 3 is the highest point along the cycle.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-16-0353.1
The isentropic streamfunction averaged over the fifth day of the simulation is shown in Fig. 4. For a steady flow, the mean flow in z–θe coordinates follows the isolines of the streamfunction. In Fig. 4, this flow would be clockwise, with air rising at high value of 
There are substantial differences between the overturning identified by the Eulerian and isentropic streamfunctions depicted respectively in Figs. 3c and 4. Notably, the mass transport is much larger in the isentropic analysis, with a maximum value of about 
2) MAFALDA trajectories
The first trajectory (solid black line) is referred here to as the inner-core cycle and is associated with air parcels rising at very high equivalent potential temperature, with 
3) Isentropic average of state variables
Isentropic-averaged value for (a) radius, (b) tangential wind, (c) specific moist entropy, (d) temperature, (e) mixing ratio, and (f) Gibbs free energy of water vapor. The solid black line and the dashed blue line correspond to Eulerian trajectories associated with the inner-core cycle and rainband cycle, respectively (see section 4).
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-16-0353.1
Figures 5c and 5d show the distribution of moist entropy 
4) State variables along the MAFALDA trajectories
We apply the MAFALDA procedure to reconstruct the thermodynamic cycles during the fifth day of our simulation. Figure 6 shows the results for the inner-core cycle and the rainband cycles under six different coordinate pairs: moist entropy s and temperature T (Fig. 6a), buoyancy b and height z (Fig. 6b), total water content 
The inner-core cycle and rainband cycle are shown in different coordinate pairs: (a) specific moist entropy and temperature, (b) buoyancy and height, (c) total water content and height, (d) Gibbs free energy of water vapor and mixing ratio, (e) liquid water content and Gibbs free energy for liquid water, and (f) ice content and Gibbs free energy for ice. The inner-core cycle is shown by the solid black line and the outer cycle by the dashed blue line. The trajectories are clockwise in all panels.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-16-0353.1
4. Thermodynamic cycles in a simulated hurricane
The MAFALDA procedure has allowed us to extract thermodynamic cycles from the numerical model output. We now turn to the physical interpretation of the cycles in various thermodynamic coordinates as shown in Fig. 6 and their implications for the generation of kinetic energy.
In the T–s diagram (Fig. 6a), the two trajectories exhibit features of a heat engine. For the inner-core cycle, the first transformation from 1 to 2 leads to an entropy increase from 200 to 300 J K−1 kg−1 due to the energy fluxes from the ocean surface. The second transformation from 2 to 3 corresponds to an expansion with approximately constant moist entropy but decreasing temperature from 300 to about 200 K. In the last leg from 3 to 1, the parcel is compressed back to the surface and its temperature increases from 200 to about 300 K. As first, the parcel loses energy and its entropy decreases from about 300 to 200 J K−1 kg−1. Closer to the surface, water vapor gained from mixing with cloudy air leads to an entropy increase from 200 to 240 J K−1 kg−1.
The rainband cycle differs from the inner-core cycle in three aspects. First, the entropy increase in the inflow portion of the cycle (1 → 2) is substantially less for the rainband cycle indicative of weaker surface energy fluxes. Second, the entropy decreases from about 280 to 250 J K−1 kg−1 during the ascent (2 → 3). This loss of entropy occurs as the air parcel loses water vapor through detrainment and mixing: a reduction of entropy of 30 J K−1 kg−1 corresponds approximatively to a loss of 3 g kg−1 of water vapor. Finally, the rainband cycle is shallower, reaching a height of 12 km and its minimum temperature (at about 220 K) is substantially warmer than for the inner-core cycle.
Figure 6c shows the two cycles in total water mixing ratio and height coordinates. Both cycles corresponds to a net upward transport of water in all phases. The geopotential energy gained by the water as it is lifted by atmospheric motions is proportional to the area within the cycle. The inner-core cycle does more work in order to lift more water to a higher level than the rainband cycle. The maximum mixing ratio in the inner-core cycle is about 22 g kg−1, which is about 2g−kg−1 larger than for the outer rainband cycle. This is consistent with the difference of about 20 J K−1 kg−1 in the maximum entropy between the two cycles and confirms that the entropy increase near the center of the storm is due to the enhanced evaporation from the ocean.
These cycles differ from a Carnot cycle in a more fundamental way: most of the entropy increase arises from the evaporation of water at the ocean surface. The air parcel must be treated as an open system that exchanges water in various phases. Figure 6d shows the two cycles in rυ–gυ coordinates with clockwise trajectories. The Gibbs free energy of water vapor can be approximated as 
Figures 6e and 6f show the two cycles in the mixing ratio and Gibbs free energy for liquid water (gl–rl in Fig. 6e) and ice (gi–ri in Fig. 6f). These are necessary for the computation of the Gibbs penalty 
For the rainband cycle, our analysis yields an external heating Qin =19.9 kJ g−1 occurring at an average temperature 
In contrast, the inner-core cycle is associated with a larger energy transport, with a net heating of 
Our analysis indicates that a striking sixfold increase in kinetic energy generation between the rainband cycle and the inner-core cycle is due to a combination of three changes: 1) a 60% increase in the external heating associated with the intense evaporation at the center of the storm, 2) a substantial decrease in the cooling temperature (from 269 to 233 K) that results in a doubling of the Carnot efficiency, and 3) the actual efficiency of the cycle becomes close to its Carnot efficiency. This later point can be attributed to the fact that relative increases in water lifting 
The increase in surface heating between the rainband and inner-core cycles is a consequence of the enhanced surface evaporation near the storm center, which has long been recognized as one of the key requirements for the maintenance of hurricane. Enhanced evaporation by itself may not be sufficient however. Indeed, the maximum intensity theory of Emanuel (1986) shows that the maximum wind depends not on entropy itself but on the entropy gradient near the storm center. To be effective, surface evaporation must lead to a local increase in the moist entropy. The ratio 
The reduction of cooling temperature 
Finally, the high generation rate of kinetic energy in the inner-core cycle is due in part to the fact that this cycle is able to achieve an efficiency that is close to the Carnot efficiency. While both the Gibbs penalty 
We further analyze 20 cycles from MAFALDA, ordered from the deepest inner-core cycle 1 to the shallowest cycle 20, with the rainband cycle described above corresponding to cycle 8. The cycles are constructed from different values of the streamfunction and are ordered from the deepest to the shallowest. Figure 7a shows the four terms from Eq. (10). Deep cycles transport more energy across a larger temperature difference and are associated with large value of the maximum work 
Thermodynamic analysis for 20 MAFALDA cycles. (a) Decomposition of the maximum work 
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-16-0353.1
Figure 7b compares the actual efficiency to the Carnot efficiency for each cycle. Deep cycles not only exhibit a higher Carnot efficiency, but they achieve an actual efficiency close to its theoretical maximum. This indicates that, while the hydrological cycle acts to greatly reduce the kinetic energy output of shallow convection, it only marginally reduces the output of deep overturning flows such as the inner-core cycle. Finally, Fig. 7c shows the temperature of the heat source 
Figure 8 shows the evolution of the thermodynamic properties of the deepest MAFALDA cycle through the 15 days of our simulation. This cycle is associated with the value of the isentropic streamfunction equal to 2.5% of its absolute minimum, which corresponds to the inner-core cycle discussed earlier. The four terms of the kinetic energy budget (10) are shown in Fig. 8a. Both the Gibbs penalty 
Time evolution of the MAFALDA cycle associated 2.5th percentile of the isentropic streamfunction, which corresponds to the inner-core cycle discussed in section 4. (a) Decomposition of the maximum work 
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-16-0353.1
5. Conclusions
In this paper, we have applied MAFALDA to analyze the thermodynamic transformations in a high-resolution simulation of a hurricane. This technique relies on identifying the atmospheric overturning by computing a mean circulation in z–θe coordinates and extracting a set of thermodynamic cycles that represent the mean overturning flow. This then allows us to diagnose various thermodynamic transformations that occur through each cycle.
We use MAFALDA here to assess the ability of the hurricane to act as a heat engine. Previous studies (Pauluis and Held 2002a,b; Pauluis 2016; Laliberté et al. 2015) have demonstrated that the hydrological cycle has a negative impact on the ability of the atmosphere to generate kinetic energy. This arises from two key aspects of the hydrological cycle. First, mechanical work must be performed in order to lift water and is then lost through frictional dissipation as condensed precipitates (Pauluis et al. 2000). Second, the atmosphere acts partially as a dehumidifier, in which water is introduced as unsaturated water vapor and removed as a condensate. From a thermodynamic point of view, the water has a lower Gibbs free energy when it enters the atmosphere than when it is removed. This results in a reduction of the amount of work that can be produced by the atmospheric circulation (Pauluis 2011). For moist convection, previous studies (Pauluis and Held 2002a; Pauluis 2016) have found that the generation of kinetic energy of moist convection in radiative convective equilibrium is about 10%–20% of the work that could be done by a Carnot cycle acting between the same energy sources and sinks.
Here, we contrast two thermodynamic cycles associated with different trajectories in our simulation: a rainband cycle associated with air ascending in the outer rainband located about 200 km away from the storm and an inner-core cycle corresponding to air rising within the eyewall. These two cycles exhibit very different thermodynamic behavior, and, in particular, the generation of kinetic energy for the inner-core cycle is approximately 6 times larger than for the rainband cycle. We identify three different factors contributing to the high generation rate of the inner-core cycle: 1) an enhancement of the energy transport by the cycle; 2) a very low cooling temperature, characteristic of the upper troposphere, which results in a very high Carnot efficiency; and 3) a relatively small negative contribution from the hydrological cycle, so that the actual efficiency of the inner-core cycle is about two-thirds of its Carnot efficiency.
The high rate of generation of kinetic energy in the inner-core cycle is strongly tied to the nature of the rising motions within the eyewall. The ascent in the rainband cycle shows a clear indication of entrainment as a gradual decrease of entropy and equivalent potential temperature as the air rises. In contrast, the ascent in the inner-core cycle shows little indication of entrainment of dry air and is almost isentropic. The ascent in the inner-core cycle reaches very high and is associated with very low cooling temperature, which greatly increases the Carnot efficiency. In our simulation, a drop in cooling temperature and a corresponding increase in efficiency precede intensification by about one day. While our work here is limited to a single storm, it strongly suggests that entrainment of dry air into the eyewall, or rather the lack thereof, plays an important role in the intensification and energetics of tropical cyclones.
The methodology of MAFALDA is designed to analyze the thermodynamic processes in a numerical simulation. The physical insights it provides should be tempered by the fact that a numerical simulation is at best a good faith effort at reproducing a physical flow. In particular, the horizontal resolution of 2 km here is too coarse to fully capture the turbulent nature of entrainment. While we strongly believe that the results presented here are both physically consistent and robust, understanding how numerical resolution and the various physical parameterizations affect the behavior of simulated atmospheric flows remains an important challenge in atmospheric science. Assessing thermodynamic processes represented in such numerical simulations should be an essential component of such an endeavor.
The novel approach introduced in this study offers a unique perspective on the role played by thermodynamic processes in hurricane formation and intensity. Our study indicates that the atmospheric circulation in a hurricane, characterized by very high generation of kinetic energy, is in a different thermodynamic regime than tropical deep convection. The genesis and intensification of tropical cyclones correspond to the emergence of deep and highly efficient thermodynamic cycles. Systematic applications of MAFALDA should shed further light on how such cycles emerge, and how energy exchanges with both the ocean surface and the surrounding environment can impact the storm intensity and structure, and on how hurricanes and tropical storms behave under different climates.
Acknowledgments
Olivier Pauluis is supported by the New York University Abu Dhabi Research Institute under Grant G1102. Fuqing Zhang is partially supported by ONR Grant N00014-15-1-2298. The WRF simulation is performed and archived at the Texas Advanced Computing Center. We thank Dandan Tao for running the numerical simulation used in this study. We benefited from discussions with Kerry Emanuel and Juan Fang.
APPENDIX
Gibbs Relationship for Moist Air
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