## 1. Introduction

Tropical cyclones (TCs) are powered by surface enthalpy fluxes (e.g., Emanuel 1986), with simulations of dry TCs (Mrowiec et al. 2011; Cronin and Chavas 2019, hereafter CC19; Wang and Lin 2020, hereafter WL20), proving that surface fluxes can power a TC even in the absence of latent heating. The theoretical potential intensity (PI), an upper bound of the maximum gradient wind, of a TC has been well formulated (Emanuel 1986, 1988, 1991; Bister and Emanuel 1998; Bryan and Rotunno 2009; Emanuel and Rotunno 2011, hereafter ER11; Rousseau-Rizzi and Emanuel 2019). Theoretical models of its size and radial profile of tangential wind are also established (Emanuel 1986; ER11; Chavas et al. 2015, hereafter C15; Chavas and Lin 2016). Unlike PI, the current wind profile models cannot be solely determined by environmental parameters. Internal TC parameters, either the radius of maximum wind or the outer radius where winds vanish need to be known first.

Specifically, C15 developed a complete TC wind profile model that compares well with observed wind profiles and captures their variability (C15; Chavas and Lin 2016). The C15 model incorporates the inner-core model of ER11, based essentially on the eyewall slantwise neutrality, and the outer-region model of Emanuel (2004, hereafter E04). The outer-region boundary layer absolute angular momentum budget was connected to subsidence velocity induced by radiative cooling in E04. Wind profiles in the C15 model are constrained by several environmental and TC structure parameters, such as Coriolis parameter, maximum wind, outer radius, subsidence velocity, and surface exchange coefficients for momentum and enthalpy.

An integral constraint of surface wind profile for the whole TC, although has not been explicitly explored, might exist if the TC is treated as an atmospheric heat engine with its efficiency regulating the ratio of total surface anemonal dissipation to total heat input (Emanuel 1997, 2003; Bister et al. 2011; Rousseau-Rizzi and Emanuel 2019). Such an approach may complement the C15 model, which has not fully accounted for the thermodynamic properties of TCs. On the other hand, the C15 model will also benefit from the heat-engine approach in the explanation of variability of the integral property of the wind field. One of the reasons why the heat-engine-based surface wind structure constraint has not been explicitly explored may be related to the relatively moderate change of nondimensional wind profiles in the current climate (Chavas and Emanuel 2014). However, recent studies have shown significant structural changes of TCs as the environment changes from dry to moist and from cold to warm (CC19; WL20). The structural change of TC surface wind profile in CC19 would be seen as a direct response to the changes of thermodynamic properties of the environment, as implicitly suggested in WL20. This provides a way to explicitly explore the surface-wind-structure dependence on the heat-engine properties of the TC.

As two extreme types of TCs in CC19, the dry and its equivalent moist counterpart, moist reversible TCs where condensates do not fall out (WL20), show significantly different structures from a typical moist TC (Mrowiec et al. 2011; CC19; WL20). Specifically, the dry and moist reversible TCs have a much larger inner-core size relative to the outer radius with an order-of-magnitude larger subsidence velocity, along with a much deeper (~4 km) inflow layer and the absence of a lower tropospheric low-entropy layer in the subsidence region. The semidry and semiwarm TCs mainly show structures in between (CC19). As discussed in CC19 and WL20, the larger inner-core size of dry and moist reversible TCs is a consequence of the order-of-magnitude larger subsidence velocity in the outer region explained by the C15 model.

However, the significant thermodynamic difference between a typical TC and a dry or moist reversible TC has not been explicitly accounted for in the C15 model. Among those parameters in the C15 model, only the subsidence velocity reflects different thermodynamic processes between TCs. In fact, the structural difference between a typical TC and a dry or moist reversible TC may be more fundamentally controlled by different heat engine efficiencies. As noted in WL20, the mechanical efficiency (ratio of mechanical work to energy input) of a typical moist TC may be significantly smaller than that of dry and moist reversible TCs, as suggested by the much larger wind fields and vertical mass fluxes than a typical moist TC. This may lead to a much more compact inner core of the typical moist TC than a dry or moist reversible TC.

In general, a heat engine expression for a climate or atmospheric system may be derived by combining an energy and an entropy budget (e.g., Bannon 2015). If no external work is done by the system, an efficiency of the system can be estimated by its irreversible entropy production (e.g., Bannon 2015). For an atmospheric system, the irreversible entropy production is due to anemonal dissipation from viscosity, precipitation dissipation resulted from the friction between falling hydrometeors and surrounding air, diffusion of sensible heat and water vapor, and irreversible phase changes (Pauluis and Held 2002a,b; Romps 2008). Of those processes, the sum of anemonal dissipation and precipitation dissipation equals to the mechanical work performed in the atmosphere in a steady state (Pauluis and Held 2002a). It was demonstrated (Pauluis and Held 2002a,b) that irreversible moist processes of diffusion of water vapor and irreversible phase changes significantly reduces the mechanical work performed in a moist atmosphere than a dry atmosphere via the corresponding irreversible entropy production. In addition, the mechanical work used to generate wind is further reduced since a substantial part of the total mechanical work is used to “lift water,” which is manifested by a source of irreversible entropy production as precipitation dissipation (Pauluis et al. 2000; Pauluis and Held 2002a,b; Romps 2008; Singh and O’Gorman 2016; Pauluis and Zhang 2017). Compared to a typical moist TC, there is no irreversible entropy production due to those processes associated with water in a dry TC, and irreversible phase changes and precipitation dissipation essentially do not exist in a moist reversible TC. Thus, it is reasonable that dry and moist reversible TCs have higher efficiencies to sustain a relatively larger inner core than a typical moist TC.

Tropical cyclones have been treated as Carnot heat engines with high efficiencies due to deep eyewall convection, but most previous studies focus on storm intensity instead of structure (Emanuel 1986, 1991, 1997; Bister et al. 2011; Sabuwala et al. 2015; Pauluis and Zhang 2017; Rousseau-Rizzi and Emanuel 2019). In this study, we will explore how a heat engine efficiency would possibly modulate the surface wind structure as manifested in dry and moist reversible TCs. The heat engine efficiency will be related to a “degree of irreversibility” *α*_{irr} (section 2) of the atmospheric system. The heat-engine modulation of surface wind structure is then further manifested by changes of the C15 model parameters. For simplicity, we will not include ice physics in the analysis as a significant structural difference between a typical TC and a dry or moist reversible TC already exists without ice physics (WL20). From a heat engine aspect, the melting cycle, in which supercooled water freezes or solid hydrometeors melts, is also a source of irreversible entropy production (Romps 2008), but not a dominant one (Singh and O’Gorman 2016).

In the following, a heat-engine-based parameter *α* representing “relative compactness” of surface wind structure will be derived and then linked to the C15 model parameters in section 2. It will be shown that *α* is modulated by *α*_{irr}, a degree of irreversibility directly determined by various irreversible entropy productions of the system. The role of *α*_{irr} in modulating *α* will be demonstrated via idealized simulations of a typical moist TC, a dry TC, and a moist reversible TC. It will be shown that *α*_{irr} is crucial for the structural difference of surface wind in the three simulated TCs. A summary and discussion of the findings is given in section 4.

## 2. Derivation of a heat-engine-based relative-compactness parameter for surface wind structure

### a. Energy and entropy budgets

*Q*

_{lat}is the surface latent heat flux,

*Q*

_{sen}the surface sensible heat flux,

*Q*

_{rad}the radiative cooling, and

*F*

_{Q}the heat flux associated with surface fluxes of water vapor and liquid water occurring at slightly different surface air temperatures, with

*L*

_{υ}the latent heat of vaporization;

*J*

^{z}the surface fluxes of water vapor, liquid water (rainfall), and sensible heat per unit area, respectively;

*c*

_{l}the specific heat of liquid water;

*T*the temperature; and

*T*

_{0}a reference temperature set to 273.15 K.

*Q*

_{lat}+

*Q*

_{sen})/

*T*

_{s}is the external entropy source by surface latent and sensible heat fluxes with

*T*

_{s}the effective temperature at which the fluxes occur;

*Q*

_{rad}/

*T*

_{rad}the external entropy sink by radiative cooling with

*T*

_{rad}the effective temperature of

*Q*

_{rad};

*F*

_{s}external entropy source related to

*F*

_{Q};

*δS*

_{dif},

*δS*

_{dv}, and

*δS*

_{pc}the entropy sources by diffusion of sensible heat, water vapor, and irreversible phase changes, respectively. The terms with

*δS*indicates internal entropy production from irreversible processes. The detailed definition of each term are as follows (Pauluis and Held 2002a,b; Romps 2008):

*d*

_{a}is the anemonal dissipation per unit volume,

*d*

_{p}is the precipitation dissipation per unit volume,

**g**the gravitational acceleration,

**d**

_{l}the diffusion flux of liquid water including rainfall,

**d**

_{υ}the diffusion flux of water vapor,

**J**the diffusion flux of sensible heat,

*e*the partial pressure of water vapor,

*R*

_{υ}the gas constant of water vapor,

*T*

_{s}as

*T*

_{rad}is also defined accordingly. For the following presentation, it is also convenient to define the effective temperature of anemonal dissipation

### b. A heat-engine-based relative-compactness parameter α

*ϵ*

_{C}= (

*T*

_{s}−

*T*

_{rad})/

*T*

_{s}is the Carnot efficiency,

*β*

_{F}= (

*F*

_{s}−

*F*

_{Q}/

*T*

_{rad})/(

*Q*

_{sen}+

*Q*

_{lat}) and

*β*

_{na}=

*δS*

_{na}/(

*Q*

_{sen}+

*Q*

_{rad}) are ratios with a dimension of

*T*

^{−1}. Similar to Lucarini et al. (2010, 2011), a “degree of irreversibility” can be defined:

*α*

_{irr}ranges from 0 to 1 and a higher value indicates a higher degree of irreversibility of the system. With Eq. (5), Eq. (4) can be rewritten:

*α*

_{irr}would decrease the anemonal-dissipation efficiency (which can be defined as

*ϵ*

_{a}=

*D*

_{a}/(

*Q*

_{sen}+

*Q*

_{lat})) of the system, given other parameters unchanged.

*C*

_{D}the surface exchange coefficient of momentum,

*ρ*

_{s}the surface air density,

**V**the surface total wind,

*r*the radius, and

*r*

_{0}the outer radius of the TC.

*D*

_{a}can be related to surface dissipation by defining a ratio

*γ*:

**V**

_{m}| so that

*r*

_{0}so that

*D*

_{a}, neglecting variations of

*ρ*

_{s}:

*A*

_{D}is the effective area of

*C*

_{k}the surface exchange coefficient of enthalpy,

*T*

_{SST}, and

*k*

_{s}the enthalpy of surface air. Similarly, the air–sea enthalpy disequilibrium is nondimensionalized by its value at the

*r*

_{m}, radius of |

**V**

_{m}|, so that

*m*denotes evaluation at

*r*

_{m}. In a similar manner, we can rewrite

*Q*

_{lat}+

*Q*

_{sen}:

*A*

_{h}is the effective area of surface heat fluxes:

*A*

_{h}should be interpreted as the area where boundary layer entropy increases radially inwards.

*ϵ*

_{PI}= (

*T*

_{s}−

*T*

_{o})/

*T*

_{o}is a highly idealized anemonal-dissipation efficiency, with

*T*

_{o}the outflow temperature, close to a tropopause temperature. Note that

*ϵ*

_{PI}can be significantly larger than

*ϵ*

_{C}because

*T*

_{o}is much smaller than

*T*

_{rad}(see section 3).

**V**

_{m}| = |

**V**

_{m–PI}| gives

**V**

_{m}| and

*γ*by controlling maximum surface wind speed.

We should also note that *α* is not an absolute radius as a size parameter. Instead, it quantifies the normalized variation of the surface wind speed *α* is expected to be mainly modulated by *α* to be much larger in a compact TC with relatively strong wind (compared to |**V**_{m}|) covering a small area, than a broad TC with relatively strong wind covering a large area. In this manner, we may interpret *α* as the ratio of a storm that is covered by relatively strong surface winds, with higher *α* values indicating a smaller fraction.

Overall, the heat-engine-based TC surface wind structure parameter *α* emerges from the following reasoning. 1) Heat engine variables, e.g., atmospheric degree of irreversibility, modulate the anemonal dissipation efficiency; 2) in a TC, the surface anemonal dissipation is assumed to take a considerable portion of the total anemonal dissipation in the atmosphere; 3) surface anemonal dissipation efficiency is modulated by surface wind structure because surface enthalpy flux and surface anemonal dissipation vary differently with surface wind speed (first power versus third power of surface wind speed). Thus, heat engine variables are linked to TC surface wind structure.

### c. Response of α to C15 model variability

It is noted that *α* defined above [Eq. (13)] applies to any axisymmetric flow, not exclusively to a TC and it is difficult to explain a given change of *α* as it is defined by a ratio of two integrals associated with the wind profile. However, if the flow is a TC, the C15 model has an advantage that the wind profile can be determined by only a few parameters. Thus, a change of *α* can be a result of a joint or individual changes of C15 model parameters. This would give a clear practical meaning of *α* when it is applied to a TC. Thus, we adopt the complete model of TC radial wind profile of C15 and show how *α* would respond to a change of parameters in C15. As a simplification, the role of

*w*

_{cool}is the subsidence velocity (positive downward). Note that Eq. (14) is not the original ER11 analytical solution because ER11 assumed that the absolute angular momentum

*M*=

*rυ*by neglecting the Coriolis parameter

*f*. Thus, Eq. (14) does not guarantee that

**V**

_{m}|/(

*fr*

_{0}),

*C*

_{k}/

*C*

_{D}, and

**V**

_{m}|/(

*fr*

_{0}) and

*C*

_{D}|

**V**

_{m}|/

*w*

_{cool}. A complete wind profile

**V**

_{m}|/(

*fr*

_{0}),

*C*

_{k}/

*C*

_{D},

*C*

_{D}|

**V**

_{m}|/

*w*

_{cool}need to be externally given.

To obtain wind profiles from C15 model, it is convenient to externally set |**V**_{m}|/(*fr*_{0}), *C*_{k}/*C*_{D}, *C*_{D}|**V**_{m}|/*w*_{cool} to be internally determined. Thus, ER11 model is first specified and then E04 model is appended to it. Reference values of these parameters in current climate are *C*_{k} = *C*_{D} = 0.001 25, giving *C*_{k}/*C*_{D} = 1, *f* = 0.000 05 s^{−1}, |**V**_{m}| = 70 m s^{−1}, and *r*_{0} = 900 km, giving |**V**_{m}|/(*fr*_{0}) = 1.56. With other parameters set to reference values, when *w*_{cool} increases from 0.2 to 2.9 cm s^{−1} as *α* decreases from 21.1 to 3.8. This result is consistent with the expectation above that a broader inner-core results in a smaller *α*. Notably, the sensitivity of *α* to

Next, |**V**_{m}|/( *fr*_{0}) is varied from 0.5 to 2.6 with other parameters at reference values. An increase of |**V**_{m}|/( *fr*_{0}) mainly results in a weakening of outer-region wind profile but a strengthening of inner-core wind profile (Fig. 2a). As a compensating result, a larger |**V**_{m}|/( *fr*_{0}) (0.5–2.6) results in a smaller *α* (14.5–12.8) (Fig. 2b). The corresponding *C*_{D}|**V**_{m}|/*w*_{cool} varies from 61.2 to 16.7. Assuming *fr*_{0} unchanged, *w*_{cool} would increase from 0.05 to 0.9 cm s^{−1}; Assuming |**V**_{m}| unchanged, *w*_{cool} would increase from 0.1 to 0.5 cm s^{−1}. It is also noted that the sensitivity of *α* to **V**_{m}|/( *fr*_{0}).

Finally, *C*_{k}/*C*_{D} is varied from 0.125 to 1.0 with other parameters set to reference values. Changes of *C*_{k}/*C*_{D} affect both inner-core and outer-region wind profiles (Fig. 3a). An increase of *C*_{k}/*C*_{D} results in an overall weakening of the wind profile, with the corresponding *α* increasing from 6.9 to 13.5. If *C*_{k} is assumed constant, *w*_{cool} would decrease from 7.5 to 0.3 cm s^{−1}; otherwise *w*_{cool} would decrease from 0.9 to 0.3 cm s^{−1}. It is cautioned that varying *C*_{k}/*C*_{D} while holding **V**_{m}|/(*fr*_{0}) constant may not be physically possible. However, as observations of *C*_{k}/*C*_{D} in higher wind speeds are limited (Nystrom et al. 2020a,b), testing the response of *α* from C15 model to *C*_{k}/*C*_{D} is still potentially helpful in understanding TC structure.

The response of *α* and *w*_{cool} to a joint change of **V**_{m}|/( *fr*_{0}), and *C*_{k}/*C*_{D} is given in Fig. 4. Overall, it is evident that *α* is strongly sensitive to **V**_{m}|/( *fr*_{0}) (Figs. 4a,b). In this manner, *α* represents the compactness of wind profile relative to the outer radius; thus, *α* may be termed as a relative-compactness parameter. In addition, a decrease of *C*_{k}/*C*_{D} by increasing *C*_{D} results in a slightly smaller *α* (Figs. 4a,b). The corresponding *w*_{cool} generally decreases with increasing *α* (Figs. 4c–f) either assuming |**V**_{m}| constant or *fr*_{0} constant.

Furthermore, C15 model links the thermodynamically derived parameter *α* and a dynamical parameter *w*_{cool}. Compared to the complexity of processes involved in *α*, *w*_{cool} may be more simply determined by a radiative cooling rate and a stability of the subsidence region (E04; CC19; WL20). Given the increased atmospheric stability with warming (Sugi and Yoshimura 2012; Sugi et al. 2015), we might expect *w*_{cool} to decrease. A consequence might be an increase of *α*, associated with a decrease of *α* to it. This indicates TC might become more compact with warming.

It is hypothesized that the irreversible entropy production through irreversible moist processes yields a much larger *α*_{irr} (thus smaller anemonal-dissipation efficiency) in a typical moist TC than in a dry or moist reversible TC. Thus, *α* would be much larger in a typical moist TC, assuming the variation of other parameters in Eq. (12) not dominant. This would result in a more compact inner core of a typical moist TC than a dry or moist reversible TC as possibly be reflected by a response of

### d. Expression for |**V**_{m}|

**V**

_{m}| = |

**V**

_{m–PI}| as done in deriving Eq. (12), an expression for |

**V**

_{m}| may be given. Substituting Eqs. (7) and (9) into Eq. (6) gives

**V**

_{m}| to surface wind structure

*α*, degree of irreversibility

*α*

_{irr}and air–sea enthalpy disequilibrium at

*r*

_{m}. Equation (16) would reduce to Eq. (11) if

*γ*= 1,

*T*

_{a}=

*T*

_{s},

*α*

_{irr}= 0,

*T*

_{rad}=

*T*

_{o},

*β*

_{F}= 0, and

*α*= 1. Note that Eqs. (11) and (16) are independent as they are derived based on different sets of assumptions, such as slantwise neutrality in Eq. (11) and RCE state in Eq. (16). Equation (11) is simple because it does not take into account of the surface wind structure as Eq. (16). To some extent, Eq. (16) may be considered as a more generalized formula than Eq. (11). Note that Eq. (16) does not explicitly require

*T*

_{o}, compared to Eq. (11), though

*ϵ*

_{C}is expected to covary with

*T*

_{o}. Equation (16) indicates that both surface wind structure (

*α*) and degree of irreversibility (

*α*

_{irr}) affect TC intensity. Note that a change of

*α*

_{irr}may lead to a change of

*α*, as will be seen in steady-state TCs in section 3; but a change of

*α*does not necessarily lead to a change of

*α*

_{irr}, as will be seen in intensifying TCs in section 3. Overall, Eq. (16) states that |

**V**

_{m}| cannot be completely isolated from TC structure and that it is only one feature of the whole TC heat engine. Detailed analysis of the relation between |

**V**

_{m}| and TC structure is out of the scope of this study.

### e. Expression for α in numerical simulations

*δE*

_{res}and entropy

*δS*

_{res}are added to the lhs of Eqs. (1) and (3), respectively. On the other hand, |

**V**

_{m}| may differ from |

**V**

_{m−PI}|. Thus, a coefficient

*k*

_{V}= |

**V**

_{m}|

^{2}/|

**V**

_{m−PI}|

^{2}can be introduced. Similarly, surface heat fluxes in a numerical model may differ slightly (~10%) from the conventional form mentioned above. Thus, the surface heat fluxes are calculated using model parameterized surface fluxes. With these introduced parameters, we can rederive Eq. (12):

*β*

_{res}= (

*δS*

_{res}−

*δE*

_{res}/

*T*

_{rad})/(

*Q*

_{sen}+

*Q*

_{lat}). Equation (17) is the form of definition of

*α*in a numerical simulation used in the following. Note that simulated TCs need not be axisymmetric as assumed above for simplicity; thus, all integral quantities for numerical simulation are calculated without assuming axisymmetry. Correspondingly, |

**V**

_{m}|

^{2}and

## 3. Numerical simulations

In this section, we test the role of degree of irreversibility, determined by the entropy production of irreversible processes, in modulating TC surface wind field. As indicated in section 2, the nondimensional surface wind structure of a TC should be modulated by the degree of irreversibility, with a higher degree of irreversibility corresponding to a more compact surface wind structure, assuming other parameters unchanged. In this section, we explicitly test this logic by numerical simulations. For simplicity and clarity, we choose a typical moist TC (CTL), a dry TC (DRY) and a moist reversible TC (REV) for the test, because the degree of irreversibility would be in sharp contrast among them. Simulation setup is identical to WL20, but with a focus on the RCE states corresponding to the derivation in section 2. As we focus on the role of degree of irreversibility, we would need to provide similar Carnot efficiencies for the three TCs by specifying certain initial sounding for each TC. Note that Carnot efficiency and PI efficiency may also modulate TC surface wind structure as seen in Eq. (12), and they are discussed in section 4.

### a. Model configuration

The model used is Cloud Model 1 (CM1) release 19.2 of Bryan and Fritsch (2002). CM1 is a three-dimensional, nonhydrostatic numerical model mainly designed for idealized simulations. The domain is set relatively small (1600 × 1600 km^{2}) to efficiently obtain a global steady state of energy and moisture in relatively short integration time, but large enough to encompass a typical TC. The horizontal resolution is 4 km throughout the domain. The domain height is ~25 km with the smallest vertical grid spacing of 50 m in the lowest 200 m of altitude and gradually increasing to 1200 m at ~20 km of altitude and remaining the same upward. The lateral boundaries are rigid walls. Rayleigh damping with an *e*-folding time of 300 s is applied from 100 km to the lateral boundaries and above 22 km of height to reduce the gravity waves reflected from the boundaries as well as to provide angular momentum sources at the boundaries.

For the CTL simulation, the microphysics scheme used is the Kessler liquid-only scheme (Kessler 1995) for simplicity of energy and entropy calculations and discussion. The base-state sounding used is from Dunion (2011) and *T*_{SST} is set constant at 303.15 K.

The initial environment for DRY is set following WL20. The base-state sounding is dry adiabatic below a tropopause at 7.7 km of altitude, giving a tropopause temperature of about 220 K. The *T*_{SST} is set to 311.15 K and the surface air temperature is set to be 15 K lower than *T*_{SST} to provide enough surface enthalpy fluxes (Mrowiec et al. 2011; WL20). These settings of base state are also aimed to generate a similar *ϵ*_{C} in the CTL and DRY. No water vapor or surface flux of water vapor is included in the DRY.

The setting for REV simulation also follows WL20. The base-state sounding is a reversible moist adiabatic below 16 km with all condensed water left in the air to represent the troposphere. *T*_{SST} is 303.15 K and the air–sea temperature difference is set to 5 K to power a reversible moist TC. The microphysics scheme is also the Kessler scheme but with autoconversion turned off so that the condensed water will follow the parcel. An artificial diffusion of liquid water to the sea is added by capping the liquid water mixing ratio at the lowest model level at 0.001 kg kg^{−1} (WL20). Note that the buoyancy frequency used in calculating subgrid turbulence is modified to be calculated as if the air were not saturated [Eq. (15) of Bryan and Rotunno 2009], in order to prevent substantial spurious heating from parameterized potential temperature turbulent fluxes.

The following configurations are the same for CTL, DRY, and REV. The radiation is represented by a Newtonian cooling capped at 2 K day^{−1} following Rotunno and Emanuel (1987). The surface exchange coefficient for momentum *C*_{D} and enthalpy *C*_{k} are set constant at 1.5 × 10^{−3}. The horizontal and vertical mixing length *l*_{h} and *l*_{υ} for the subgrid-scale turbulence are set 1500 and 100 m, respectively. Dissipation of kinetic energy from subgrid turbulence parameterization is converted to internal energy of the air for the conservation of energy. Dissipative heating from falling hydrometeors is also turned on. The diffusion of sensible heat between falling hydrometeors and air is also included. The initial vortex is the same as Rotunno and Emanuel (1987) with the maximum tangential wind speed of 12 m s^{−1} at about 100 km of radius. The simulation is performed for 40 days to achieve a relatively steady state of energy and moisture for the simulated TCs.

### b. Simulated TC evolution

The evolutions of surface horizontal wind speed and air–sea enthalpy disequilibrium *A*_{D} covers the strong-wind areas, while *A*_{h} expands over the weak-wind areas.

To test the hypothesis in section 2c, we should ensure that the energy of the system has reached a quasi-steady state. The time series of perturbation of total energy (sum of internal, potential, and kinetic energies), and of the perturbation of total water mass integrated in the domain for CTL, DRY, and REV are shown in Fig. 6. The total energy perturbation in CTL undergoes a rapid decrease from 0 to about 200 h (Fig. 6a), associated with the loss of water mass in the same period (Fig. 6b). The total energy then increases and becomes quasi steady afterward (Fig. 6a) with a slight decreasing trend after 800 h. The total water mass perturbation (Fig. 6b) is always decreasing till ~400 h. The decrease of water mass is consistent with the dehumidifying property of moist convection (Pauluis and Held 2002a,b). Total energy and water mass are quasi steady in the period of 550–800 h as well as the surface wind and air–sea enthalpy disequilibrium (Figs. 5a,b). Thus, this period is reasonably taken as the steady state to be analyzed.

The perturbation of total energy in DRY becomes quasi steady after 400 h (Fig. 6c). Given that the surface wind and air–sea enthalpy disequilibrium are also quasi steady after ~400 h, we choose the same period of 550–800 h as CTL for the steady state. The perturbation of total energy in REV approaches a quasi-steady state after ~600 h and the total water mass after ~800 h with a small preceding increasing tendency. Given that the evolution of the surface wind and air–sea enthalpy disequilibrium are also of a similar trend, we choose the period of 800–950 h as the steady state.

### c. Steady-state energy and entropy budgets and α

The total input energy averaged over the whole domain during the steady state is ~150 and 165 W m^{−2} in CTL and REV, respectively, dominated by latent heat flux, while it is ~130 W m^{−2} in DRY (Table 1). Radiative cooling largely balances the surface heat fluxes, leaving a small residual due to small fluctuation of the system and numerical errors. The total dissipation, equal to mechanical work, *D*_{a} + *D*_{p} is much larger in DRY and REV than in CTL. This is due to a much larger mechanical efficiency [*ϵ*_{m} = *D*_{a} + *D*_{p}/(*Q*_{lat} + *Q*_{sen})] and anemonal-dissipation efficiency (*ϵ*_{a}) in DRY (0.097; 0.097) and REV (0.069; 0.069) than CTL (0.042; 0.021), consistent with the expectation discussed in section 2.

Energy budget (W m^{−2}) for CTL, DRY and REV in the steady state [Eq. (1)], with total latent heat *Q*_{lat}, sensible heat *Q*_{sen}, radiative cooling *Q*_{rad}, and the residual term *Q*_{res}.^{1} Anemonal dissipation *D*_{a} and precipitation dissipation *D*_{p} are also shown.

The entropy budgets during the steady state for three experiments are given in Table 2. The external entropy source from sea surface heat fluxes is slightly smaller than the external entropy sink from radiative cooling, with the rest (~10%) of the external entropy sink balanced by irreversible entropy production from various internal processes. Note that anemonal dissipation takes ~87% and ~79% of the total irreversible entropy production in DRY and REV, but only ~19% in CTL. Comparing CTL and DRY, the difference is mainly contributed by the three irreversible processes of diffusion of water vapor, irreversible phase changes and falling hydrometeors, which generate comparable entropy productions as anemonal dissipation in CTL but do not exist in DRY. The difference between CTL and REV is also mainly contributed by the three irreversible processes. Water vapor diffusion contribution is much smaller in REV than CTL. Irreversible phase changes and falling hydrometeors generate entropy productions in CTL but essentially do not exist in REV.

Entropy budget (m W K^{−1} m^{−2}) for CTL, DRY and REV in the steady state [Eq. (2)] with the residual term.

During the steady state of each of the three experiments, *α* is calculated using Eq. (17). The values of the associated parameters are given in Table 3. The Carnot efficiency *ϵ*_{C} are similar (~0.11) in three experiments, giving a fair comparison for *α*. *α* is ~5 times larger in CTL (18.7) than DRY (3.7) and REV (4.1). The corresponding *α*_{irr} is significantly larger in CTL (0.81) than DRY (0.13) and REV (0.21), supporting the discussion in section 2.^{1} In terms of its definition [Eq. (13)], the much larger *α* in CTL is due to the much smaller *A*_{D} associated with the compact inner core (Fig. 5).

In three experiments, *γ* is about 2, suggesting that the anemonal dissipation above the surface is comparable to that at the surface. *T*_{s} and *T*_{o} under current climate. This may result from specific model configurations, such as a large horizontal mixing length (Bryan and Rotunno 2009) and a relatively low horizontal resolution. A closer look appears to indicate that the boundary layer closure of the PI theory is violated to some extent in these simulations (not shown). The significantly larger |**V**_{m–PI}| than |**V**_{m}| does not mean that the same is true in nature and under other model configurations. It also does not impact the TC surface wind structures, which we presently focus on, as found in WL20 with higher horizontal resolution and larger domain size. The value of *T*_{rad} is about 270 K in three experiments, significantly higher than the tropopause temperature (~200 K in CTL and REV, ~220 K in DRY), while it should be cautioned that *T*_{rad} could be very sensitive to the choice of the radiative scheme, and to other model parameters. The value of *T*_{a} is about 290 K, close to the surface temperature. Compared to *ϵ*_{C}, *β*_{F}*T*_{rad} is negligible in three experiments; *β*_{res}*T*_{rad} is also generally negligible except for a relatively larger value in REV.

Because parameters other than *α*_{irr} only experience small variations or variations with limited effects, one way to evaluate the role of *α*_{irr} in determining *α* is to substitute the value of *α*_{irr} in CTL by its values in DRY and REV, or by doing the reverse.^{2} Determining *α* in CTL using *α*_{irr} in DRY and REV with other parameters unchanged, we get an *α* of 4.1 and 4.5, respectively, which is significantly smaller than the original value of 18.7 and becomes qualitatively the same as the *α* of 3.7 in DRY and 4.1 in REV. The small differences are due to other different parameters between the experiments. Similarly, determining *α* in DRY and REV using *α*_{irr} in CTL with other parameters unchanged, we get an *α* of 17.0 and 17.2, respectively, which is much closer to the *α* of 18.7 in CTL. The differences of other parameters cannot explain the difference of *α* between CTL and DRY or REV. Overall, it is evident that *α*_{irr} is a key parameter for the magnitude of *α* in the three experiments.

Consistent with section 2c, C15 model is adopted to more clearly see what size parameters are changed in response to the change of *α* among the experiments. In this process, *α*. To test this assumption, *α* is 12.4 for CTL, 2.6 for DRY, and 3.4 for REV assuming a uniform air–sea enthalpy disequilibrium (see also Fig. 9a), indicating that *α* compared to *r*_{8} (radius where wind speed equals to 8 m s^{−1} in the subsidence region) given simulated *w*_{cool}, and then fitting ER11 model to E04 model by tuning *r*_{m}. *w*_{cool} is taken as the area-weighted subsidence velocity between 2 and 5 km of altitude and 250–700 km of radius in CTL, 500–700 km of radius in DRY and REV. The resulting *w*_{cool} is 0.22, 2.6, and 2.4 cm s^{−1} for CTL, DRY, and REV, respectively. The resulting *r*_{0} of C15 model after fitting is 850 km for CTL, 781 km for DRY, and 713 km for REV.

The nondimensional surface wind profiles of CM1 and C15 model are shown in Fig. 7. Note that surface wind does not necessarily vanish at 800 km of radius in CM1 simulations because TC sizes saturate at the domain size, which is necessary for the quasi-steady state to be efficiently established. The resulting *α* in C15 model is 15.5 for CTL, 2.2 for DRY, and 2.9 for REV, assuming negligible variation of **V**_{m}|/( *fr*_{0}) in C15 model is 1.21, 0.81, and 1.12 for CTL, DRY, and REV, respectively; and the resulting *α* on a joint change of |**V**_{m}|/( *fr*_{0}) and *α* in fitted C15 model.

Knowing the importance of irreversible entropy production by various moist irreversible processes in determining *α* through *α*_{irr}, the radial distributions of irreversible entropy productions during the steady state can be checked for a better understanding of the TC energetics. First of all, in CTL, irreversible entropy production from anemonal dissipation is accumulated significantly within 100 km of radius (Fig. 8a), where wind is strongest. The precipitation dissipation accumulates more sharply within 100 km of radius, where convection and precipitation are strongest (Fig. 8a). Accumulated irreversible entropy production from irreversible phase changes, diffusion of water vapor and diffusion of sensible heat slowly increase with radius throughout the TC (Fig. 8a). Specifically, evaporation of seawater into subsaturated surface air takes ~80% of the total irreversible entropy production by irreversible phase changes. The rest from reevaporation of rain in subsaturated environment takes only ~20%, occurring in the inner-core region (not shown). The accumulated radial distribution of irreversible entropy production (Fig. 8a) shows that irreversible processes occur in the subsidence area of the TC are important in determining the TC surface wind structure in a moist atmosphere reflected by parameter *α*.

The accumulated radial distributions of irreversible entropy production in DRY and REV are shown in Figs. 8b and 8c, also indicating the importance of subsidence region. Irreversible entropy production from anemonal dissipation shows a much slower radial accumulation in DRY and REV compared to CTL, reflecting a larger inner-core size in DRY and REV. Accumulated radial distribution of irreversible entropy production from diffusion of sensible heat in DRY and REV is similar to the CTL. Irreversible entropy production from diffusion of water vapor is present is REV, showing a similar trend as that from diffusion of sensible heat.

### d. Time evolution of α and associated parameters in Eq. (17)

It is of concern how *α* varies during intensification and how Eq. (17) works before the RCE state, when the energy and entropy of the system is evolving. First, it is evident that *α* significantly increases during the intensification stage of CTL (Fig. 9a). This is associated with the expansion of wind field while at the same time increasing its overall compactness by significantly decreasing its *r*_{m} and *A*_{D} (Fig. 5a). After about 10 days, *α* reaches about 20 and fluctuates about this value afterward. In DRY and REV, *α* also approach the steady-state values near 200 h, qualitatively consistent with evolution of the wind fields. Note that *A*_{h} significantly increases in DRY and REV associated with the expansion of wind field during the intensification (Figs. 5c–f). The trend of evolution for *α* assuming *α* itself (Fig. 9a).

To further see how Eq. (17) works before the steady state, evolutions of some parameters are given. *ϵ*_{C} and *α*_{irr} rapidly reach a steady-state value within tens of hours in three experiments (Figs. 9b,c). *α*_{irr} initiates from a value close to its steady-state value. This means that *ϵ*_{C} and *α*_{irr} can be considered qualitatively independent of TC evolution. Of concern is the role *T*_{rad}*β*_{res} plays in Eq. (17) during intensification and before RCE state. This term reflects the evolution of total energy and entropy of the system. Figure 9d shows that *T*_{rad}*β*_{res} does play a role during the intensification as it may be larger (in absolute value) than the steady-state value by 0.02 during intensification (except for the first few hours in CTL). However, it becomes small as the wind field becomes quasi steady (Fig. 5). Overall, although Eqs. (12) and (17) are derived for an RCE state, the evolution of energy and entropy may not play a significant role after the TC wind field is quasi steady, and thus, Eqs. (12) and (17) may be used without special concern of the *T*_{rad}*β*_{res} term after the wind field is quasi steady.

## 4. Summary and discussion

Various irreversible processes (such as anemonal and precipitation dissipation, diffusion of water vapor and sensible heat, and irreversible phase changes), which generate internal entropy production by the second law of thermodynamics, occur in the atmosphere. In an atmospheric heat engine, the anemonal efficiency, i.e., the ratio of anemonal dissipation to surface enthalpy flux, is modulated by heat engine variables such as Carnot efficiency *ϵ*_{C}, and the degree of irreversibility *α*_{irr}, defined by the ratio of the irreversible entropy production due to processes other than anemonal dissipation to total irreversible entropy production, of the atmosphere. As surface anemonal dissipation, taking a considerable portion of the total anemonal dissipation in a TC, and surface enthalpy flux are both related to surface wind speed, this study thus links TC surface wind structure to the atmospheric heat engine variables, of which an important one is the degree of irreversibility. Specifically, as surface anemonal dissipation varies with the third power of surface wind speed while the surface enthalpy flux varies with the first power of the surface wind speed, different surface wind structure would correspond to different surface anemonal dissipation efficiency. As atmospheric heat engine properties (e.g., degree of irreversibility) change, surface anemonal-dissipation efficiency would change correspondingly, requiring a change of surface wind structure.

Specifically, a heat-engine-based relation for TC surface wind profile is derived, with assumptions of an RCE state and the validation of Emanuel’s potential intensity for TC. The relation is expressed by a “relative compactness” parameter *α*, which is defined by the ratio of effective area of surface heat fluxes to that of surface anemonal dissipation with wind speed and air–sea enthalpy disequilibrium values evaluated at the radius of maximum wind *r*_{m}. The value of *α* is modulated by the thermodynamic efficiency in the PI theory *ϵ*_{PI}, the Carnot efficiency *ϵ*_{C} of the system, and the degree of irreversibility *α*_{irr} of the system. Higher *α*_{irr} reduces the conversion efficiency of surface heat fluxes to anemonal dissipation. Specifically, higher *α*_{irr} and *ϵ*_{PI} contribute to a larger *α*, and higher *ϵ*_{C} contributes to a smaller *α*.

Assuming axisymmetric flow, *α* can also be written as an integral quantification of nondimensional radial profiles of surface wind and air–sea enthalpy disequilibrium. Change of *α* is translated to changes of the C15 model parameters, given that change of *α* is likely dominated by change of wind field rather than air–sea enthalpy disequilibrium. It is shown that *α* is strongly sensitive to *r*_{m}/*r*_{0} and weakly sensitive to |**V**_{m}|/( *fr*_{0}), assuming *C*_{k}/*C*_{D} not influenced by TC wind field. As *r*_{m}/*r*_{0} decreases, *α* increases. In this manner, *α* can be interpreted as the “compactness” of the TC inner core relative to its whole size. Furthermore, the subsidence velocity *w*_{cool} of the TC is generally found to decrease as *α* increases.

Numerical simulations are conducted to specifically reveal the role of *α*_{irr} to modulate *α* and thus surface wind structure. The modulation of *α* by *α*_{irr} is explicitly shown by idealized simulations of a typical moist TC (CTL), a dry TC (DRY), and a moist reversible TC (REV), in which hydrometeors do not fallout. The nondimensional surface wind fields in DRY and REV are much broader than CTL, as found in WL20. With certain model configurations, *ϵ*_{PI} and *ϵ*_{C} are approximately the same in CTL, DRY, and REV, among other parameters. A significantly larger *α* in CTL (18.7) than DRY (3.7) and REV (4.1) is associated with a significantly larger *α*_{irr} in CTL (0.81) than DRY (0.13) and REV (0.21). Correspondingly, the anemonal-dissipation efficiency is much higher in DRY (0.097) and REV (0.069) than CTL (0.021). The larger *α*_{irr} in CTL is contributed by the irreversible entropy production from precipitation dissipation, diffusion of water vapor and irreversible phase changes compared to DRY, where these processes are absent. Similarly, irreversible entropy production from precipitation dissipation and irreversible phase changes essentially does not exist in REV due to its experimental design. This, along with a smaller irreversible entropy production from diffusion of water vapor, contributes to the lower *α*_{irr} in REV than CTL. Irreversible entropy production from precipitation dissipation is mainly confined in the eyewall in CTL, but that from diffusion of water vapor and irreversible phase changes is contributed from the whole area of the TC, indicating the importance of the irreversible entropy production in subsidence area in TC energetics.

Though the radial variation of air–sea enthalpy disequilibrium quantitatively contributes to the large *α* in CTL, the radial variation of wind alone already produces a significantly larger *α* in CTL than DRY or REV. This suggests that *α* is mainly regulated by the wind profile rather than air–sea disequilibrium profile. Consistent with the sensitivity analysis of C15 model, the large (small) *α* produced by wind field alone in CTL (DRY or REV) is realized by a correspondingly small (large) *r*_{m}/*r*_{0}.

An analysis during the TC intensification period before the RCE state shows that *ϵ*_{C} and *α*_{irr} are qualitatively independent of TC evolution and thus may be considered environmentally determined. Evolution of total energy and entropy of the atmospheric system is not likely to play a significant role after the TC wind field is quasi steady. These results suggest that though the determination of *α* is derived assuming RCE state, it might also be applicable when the TC wind field becomes quasi steady, though an RCE state is required for quantitatively accurate results.

In this study, *α* has been interpreted as a relative compactness in terms of nondimensional wind field, with a higher value representing a more compact inner core relative to the whole size (Fig. 1a). Note that the relative compactness does not change with a uniformly proportional increase/decrease of intensity or expansion/shrink of radial extent of the wind profile.

What the effective area of surface anemonal dissipation *A*_{D} and enthalpy fluxes *A*_{h} marks has not been very explicit. As seen in Fig. 5, *A*_{D} mainly covers the strong-wind area, generally within the ascending region (not shown). However, the area covered by *A*_{h} (Fig. 5), which is well within the subsidence region of CTL but near the inner edge of the subsidence region of DRY and REV (not shown), does not appear to show a clear indication what processes may only occur inside/outside of *A*_{h}. The physical meanings of *A*_{D} and *A*_{h} still require further exploration.

From three equivalent experiments, WL20 showed that the falling nature of hydrometeors poses a strong constraint on TC structure by impacting the vertical gradient of entropy in the subsidence region. Their experiments showed that it is the subsaturation of a moist atmosphere, which is responsible for the different vertical gradient of entropy (thus stability, see also CC19) in the subsidence region, that caused the significant structural difference between CTL and DRY or REV. The significantly different wind profiles between CTL and DRY or REV are directly explained by C15 model with the order-of-magnitude-different input subsidence velocity, corresponding to different atmospheric stability. In this study, without specific attention to vertical gradient of entropy, stability, or subsidence velocity, it is further showed that in a typical moist atmosphere the subsaturation is accompanied by a high degree of irreversibility of the system that reduces the mechanical work to generate kinetic energy so that the TC surface wind field is very compact in current climate. The high degree of irreversibility ultimately results from the various irreversible moist processes in a typical moist atmosphere. Compared to WL20, this study further emphasizes the role of various irreversible processes that determine a degree of irreversibility in modulating a nondimensional surface wind structure of a TC. Combining the two studies, it could be further inferred that atmospheric stability (thus also ascent/descent-area asymmetry) is an important intermediate variable that translates the change of degree of irreversibility in response to a change of environment, to a change of subsidence velocity according to which the TC surface wind structure (compactness) can respond via C15 model.

A link is suggested between the results of Pauluis and Zhang (2017) and the present study. With the isentropic streamfunction, Pauluis and Zhang (2017) reconstructed thermodynamic cycles associated with a tropical cyclone. With the Gibbs relation, they showed theoretically that the work done to generate kinetic energy by an air parcel in each cycle is reduced by the work done to lift water and Gibbs penalty due to irreversible phase changes and water mass exchanges between the parcel and its environment. Thus, the efficiency, defined by the ratio of mechanical work to generate kinetic energy to total heat (basically latent heat) received, of a thermodynamic cycle is less than its Carnot efficiency (Pauluis and Zhang 2017). As it turns out, the eyewall cycles are very efficient, 70% of its Carnot efficiency. However, these highly efficient cycles only account for a very small fraction of the total mass transport. This would make the whole moist TC not as efficient as a Carnot heat engine, consistent with the analysis in the present work that the anemonal-dissipation efficiency of a typical moist TC is much reduced by its high degree of irreversibility. The fraction of the mass transport associated with the efficient cycles may mirror the role of “relative compactness” parameter *α* in the present study, which is a reflection of the degree of irreversibility of the whole system.

Relative-compactness parameter *α* defined in this study may share a similar property with TC fullness, defined by 1 − *r*_{m}/*r*_{17}, with *r*_{17} the radius of 17 m s^{−1} tangential wind, in Guo and Tan (2017). Guo and Tan (2017) found that TC intensity positively correlates to fullness (or relative compactness used in the present study). This is also seen in Eq. (16), which indicates a more compact TC would be more intense, given all other parameters unchanged. The intensification stage of CTL (Fig. 9a) also indicates a covariation of intensity and relative compactness (*α*). Compared to Guo and Tan (2017), the present study further provides an equation [Eq. (16)] that explicitly relates TC structure and its intensity by considering the TC as an atmospheric heat engine. Deeper understanding of intensity and structure requires further exploration.

*r*

_{m}/

*r*

_{0}) from dry to moist and from cold to warm are generally continuous and similar. As they hold

*ϵ*

_{PI}the same in their two sets of experiments, it is speculated that the gradual change of wind structure in their experiments is mainly a result of a gradual increase of

*α*

_{irr}from dry to moist and from cold to warm so that

*α*increases and

*r*

_{m}/

*r*

_{0}decreases, under an assumption that the radial variation of air–sea enthalpy disequilibrium does not have a significant contribution to the change of

*α*. In an actual warming climate,

*ϵ*

_{PI}and

*ϵ*

_{C}would both increase along with a hypothesized increase of

*α*

_{irr}because of a higher water content of the atmosphere. Though the corresponding change of

*α*needs further investigations, we might give an estimate of the change of

*α*as follows. Taking natural logarithm of Eq. (12) and differentiating the result gives

*β*

_{F}, differentiation of

*T*

_{a}/

*T*

_{rad}and

*γ*are neglected. For simplicity, we set

*T*

_{rad}=

*ηT*

_{s}+ (1 −

*η*)

*T*

_{o}, where

*η*is a coefficient. In a warming climate, we assume

*T*

_{o}is fixed and

*T*

_{s}is increased (Seeley et al. 2019). Thus, we have

*d*ln(

*ϵ*

_{PI}/

*ϵ*

_{C}) =

*d*ln

*T*

_{s}−

*d*ln(1 −

*η*). Assuming

*η*constant, we have

*d*ln(

*ϵ*

_{PI}/

*ϵ*

_{C}) =

*d*ln

*T*

_{s}> 0. With the hypothesis that

*d*ln(1 −

*α*

_{irr}) < 0 with increasing

*T*

_{s}, we have

*d*ln

*α*> 0 from Eq. (18). An increase of

*α*in a warmer climate may translate to a decrease of

*r*

_{m}/

*r*

_{0}due to the strong sensitivity of

*α*to it. Thus, this simple estimation suggests that

*r*

_{m}/

*r*

_{0}of a TC would decrease, i.e., a more compact inner core compared to the whole size, in a warmer climate. How

*r*

_{m}/

*r*

_{0}changes with

*T*

_{s}may require further numerical simulations and theoretical explorations.

## Acknowledgments

We thank Dr. Pauluis, Dr. Kowaleski, and Dr. Rousseau-Rizzi for their constructive review comments. This work was supported by the National Key Research Project of China (Grant 2018YFC1507001) and the National Natural Science Foundation of China (41975127).

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^{1}

These values of *α*_{irr} are largely determined by the environment in which the TC develops, because qualitatively the same values can be obtained from the corresponding RCE simulations without background rotation (not shown).

^{2}

This analysis also implicitly assumes that *α*_{irr} is largely environmentally determined, which is not explicitly shown, and largely independent of variations of other parameters.