1. Introduction

Increasing efforts have been given to understanding the relation between size and intensity of tropical cyclone (TC) in recent years. In particular, numerous modeling studies based on axisymmetric and three-dimensional idealized simulations found that the TC vortex initially with a larger overall (both inner core and outer core) size tends to result in a larger and more intense TC in the steady state in which the TC intensity is approximately constant (Xu and Wang 2010, 2018; Kilroy and Smith 2017; Tao et al. 2020; Li and Wang 2021; Rotunno 2022; Li et al. 2024; Fei and Wang 2024). The positive correlation between the initial and steady-state sizes reflects a strong memory of the vortex structure (Xu and Wang 2010; Tao et al. 2020; Rotunno 2022). As clarified in Rotunno (2022), based on gradient-wind theories (Emanuel 1986; Emanuel and Rotunno 2011) and boundary layer dynamics, the radius of maximum wind (RMW) or overall size in the steady state increases with the angular momentum at an outer radius and the initial RMW is the effective outer radius. Therefore, the steady-state size retains a strong memory of the initial size. However, the dynamics of how TC initial size affects its steady-state intensity has not been well understood so far. Because of the strong memory of TC size, the dependence of steady-state intensity on initial size can be investigated by examining the relation between the steady-state size and intensity.

Some recent studies have investigated the effect of steady-state size on steady-state intensity from the boundary layer dynamic perspective. Based on a series of axisymmetric numerical simulations, Tao et al. (2020) found that during the steady state the unbalanced flow, i.e., supergradient wind, in the boundary layer is stronger in a larger TC. They thus hypothesized that the contribution from supergradient wind to TC intensity increases with increasing size, which explains the increase in TC intensity with size. This hypothesis has been confirmed by Rotunno (2022), who showed that in axisymmetric numerical simulations, as the initial size and thus steady-state size increase, both the simulated and theoretical (Emanuel 1986; Emanuel and Rotunno 2011) maximum gradient wind changed a little but the maximum tangential wind in the boundary layer increased significantly due to the increase in supergradient wind. Rotunno (2022) also found that the increase in supergradient wind with the TC size is directly related to the TC boundary layer dynamics and can be well captured by the potential intensity theory in terms of boundary layer tangential winds (Bryan and Rotunno 2009).

In addition to the boundary layer dynamics, TC intensity can be also inferred from the energetics with the assumption of a TC as a heat engine. In the conceptual model of the TC heat engine (Emanuel 1997, 2018; Pauluis and Zhang 2017), the energy cycle of air parcels contains two main processes: acquiring entropy mainly by surface heating near the ocean surface at a warmer temperature and losing entropy mainly by infrared radiation in the upper troposphere at a cooler temperature. The cycle thus continuously converts heat energy into mechanical energy (mainly kinetic energy of the near-surface winds) as a Carnot heat engine. An expression of theoretical maximum intensity in terms of near-surface winds has been derived in Emanuel (1997, 2018) by rewriting the balance between production and dissipation of mechanical energy over the energy cycle to that between available production rate and dissipation rate of mechanical energy over the ocean surface at the RMW. Although the simplified model of Emanuel (1997, 2018) provides a good upper bound for the intensity of some simulated TCs, it shares a similar mathematical equation to the gradient-wind-based theoretical maximum intensity (Emanuel 1986; Emanuel and Rotunno 2011; Rousseau-Rizzi and Emanuel 2019), which has been shown to fail to capture the increasing tendency of TC intensity with size as mentioned above (Rotunno 2022). Therefore, it remains an important issue from the viewpoint of energetics as to how TC size modulates TC intensity.

As mentioned above, the main energy source of TC heat engine is surface heating by enthalpy flux (Fang et al. 2019), which depends on the near-surface wind speed. A larger overall size means a larger area coverage of high near-surface wind speed and thus surface enthalpy flux under the whole TC system (e.g., Li et al. 2020a; Li and Wang 2021). We thus hypothesize that TC size modulates the TC energy cycle and TC intensity mainly by affecting the surface enthalpy flux or the energy supply for the TC heat engine. The main purpose of this article is to test this hypothesis and to provide a new explanation of the dependence of TC intensity on size from the energetic perspective. To this end, similar to that in Tao et al. (2020) and Rotunno (2022), we conduct a series of axisymmetric simulations by varying the initial size of the TC vortex and performing detailed diagnostics mainly for the quasi-steady state. Note that to guarantee that the energy diagnostics is based on a closed thermodynamic cycle, an isentropic analysis-based energy budget (Pauluis and Mrowiec 2013; Mrowiec et al. 2016; Pauluis and Zhang 2017) is performed.

The model and experimental design are described in section 2. An overview of the simulation results is shown in section 3. Section 4 discusses the effect of TC size on the energy cycle and intensity based on the isentropic-based energy budget analysis. Main conclusions are drawn and discussed in section 5.

2. Model and experimental design

The numerical model used in this study is version 20.3 of the Cloud Model 1 (CM1; Bryan and Fritsch 2002) for axisymmetric TC simulations. The model domain is about 3000 km in the radial direction with a grid spacing of 1 km within 100-km radius and a linear stretching to 14 km at the lateral boundary. The domain is 25 km deep with stretched grid spacing from 50 to 500 m below 5.5-km height and fixed at 500 m above. A constant Coriolis parameter of f = 5 × 10⁻⁵ is used. The horizontal turbulent mixing length is set to 700 m, while the asymptotic vertical turbulent mixing length is set to 70 m. The surface enthalpy exchange coefficient C_k is fixed at 1.2 × 10⁻³. The surface drag coefficient C_D increases linearly with wind speed as wind speed is lower than 25 m s⁻¹ and is kept constant at 2.4 × 10⁻³ afterward (Donelan et al. 2004). The model physics used includes the Thompson microphysical scheme (Thompson et al. 2008) and the Newtonian cooling capped at 2 K day⁻¹ for mimicking longwave radiative cooling (Rotunno and Emanuel 1987). Cumulus convection, shortwave radiation, and dissipative heating are not included.

To ensure the robustness of the results, four sets of experiments with different sea surface temperatures (SSTs) from 28° to 31°C with 1°C increment are performed, and the corresponding atmospheric soundings are sorted for the western North Pacific (Li et al. 2020a), labeled as SST28–SST31. Note that the initial CAPE in those atmospheric soundings is nonzero and increases with increasing SST [see Fig. 8 of Li et al. (2020a)]. The radial profiles of the initial surface tangential winds are calculated based on the algorithm in Wood and White (2011) with the shape parameter of 1.6. The initial maximum wind speed is 15 m s⁻¹ for all simulations. To understand the effect of TC size on the maximum intensity, for each set of SST experiments, 21 simulations are performed with the initial RMW varied from 40 to 120 km with an increment of 4 km. By varying the initial RMW, the initial outer-core size would be also varied. All simulations are integrated for 15 days to ensure all TCs reached their quasi-steady states. Note that the 40–120-km RMW is used, as in many previous modeling studies (e.g., Xu and Wang 2010, 2018; Tao et al. 2020; Rotunno 2022; Li et al. 2024), since it corresponds to the range where the initial RMW occurs most commonly in the real atmosphere (Kimball and Mulekar 2004; Li et al. 2022; see their Figs. 1c,d). We have also examined additional simulations with larger initial RMWs up to 180 km and found that the main conclusions below are robust. However, with a larger initial size, the TC tends to have a considerably longer spinup period, especially for the simulations with lower SST. In the set of SST28, the TC cannot attain tropical storm intensity (17 m s⁻¹) until more than ∼10 days as the initial RMW is greater than ∼140 km and cannot finish intensification even after more than ∼15 days as the initial RMW is greater than ∼160 km. Considering the fact that in the real atmosphere the spinup/intensification period of TCs is generally less than 10 days (Li et al. 2022; see their Figs. 1c,d), we do not discuss the additional simulations with larger initial RMWs.

In this study, we mainly focus on the steady state of TCs in terms of intensity. We notice that most of the previous studies define the steady state as a given period near the end of simulations, such as the last 24 h of the 8-day simulations in Tao et al. (2020) and Rotunno (2022). Nevertheless, as shown in Li et al. (2024), the time in TCs attaining the quasi-steady state varies largely (∼3–10 days) with varying both SST and initial size. Previous studies have shown that in idealized simulations TCs after their steady state often further gradually evolve toward an equilibrium state that is similar to a TC attractor and mainly constrained by the boundary conditions and system properties of model (e.g., Frisius 2015). Therefore, to ensure a fair comparison, we define the steady state as the 48 h after the end of the intensification period for each TC, and all steady-state quantities discussed hereafter are defined as the average over the 48 h using 1-h output. The intensification period is defined as the period with the 24-h intensity change being continuously positive as in Li et al. (2024).

3. An overview of the simulation results

Figure 1 shows the time series of the maximum 10-m total wind speed (hereafter TC intensity), RMW, radius of 17 m s⁻¹ wind (R17), and radius of 5 m s⁻¹ wind (R5) of all simulations, in which the final 48-h period of each curve marks the defined steady state of each TC as mentioned earlier. Figure 2 shows the steady-state 10-m total wind speed profiles. As presented in previous studies based on both axisymmetric and three-dimensional idealized simulations (Xu and Wang 2010, 2018; Kilroy and Smith 2017; Tao et al. 2020; Li and Wang 2021; Rotunno 2022; Li et al. 2024; Fei and Wang 2024), the spinup period is longer and the overall size is larger throughout the spinup, intensification, and steady-state periods with a larger initial overall size (Figs. 1e–p and 2). As mentioned earlier, this reflects a strong memory of the vortex structure (Xu and Wang 2010; Tao et al. 2020; Rotunno 2022). Similar to the maximum tangential wind in the boundary layer in Tao et al. (2020) and Rotunno (2022), the steady-state TC intensity in terms of maximum 10-m total wind speed also increases as the overall size increases (Figs. 1a–d and 2). Note that the inner-core size (RMW) attains a quasi-steady state preceding the defined steady state in terms of intensity as in the real atmosphere (Li et al. 2022). With the increase in SST, the TC tends to have a shorter spinup/intensification period and higher steady-state intensity (Figs. 1 and 2), consistent with previous simulations (e.g., Li et al. 2020a). In addition, the outer-core size (R17, R5; Figs. 1i–p) tends to be larger with higher SST, corresponding to the more active convective activities in the TC outer region as analyzed in Li et al. (2020a).

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Time series of (a)–(d) the maximum 10-m wind speed, (e)–(h) the RMW, (i)–(l) the radius of 17 m s⁻¹ wind, and (m)–(p) the radius of 5 m s⁻¹ wind before and during the defined steady state for different SSTs with different colors showing the simulations with different initial RMWs (km).

Citation: Journal of the Atmospheric Sciences 82, 3; 10.1175/JAS-D-24-0042.1

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(a)–(d) Radial distributions of the steady-state 10-m total wind speed for different SSTs with different colors showing the simulations with different initial RMWs [km; color bar in (a)].

Citation: Journal of the Atmospheric Sciences 82, 3; 10.1175/JAS-D-24-0042.1

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Because of the strong memory of TC size (Figs. 1 and 2), we use the steady-state size as a proxy to understand the effect of size on steady-state intensity as in Tao et al. (2020) and Rotunno (2022), although the effect may not be instantaneous. To further identify the relation between the steady-state intensity and size, we plot the intensity as a function of the RMW, R17, and R5 (Fig. 3). The increasing tendency of the intensity can be well depicted no matter sampled by inner-core size (RMW) or outer-core size (R17 or R5). On average, the TC intensity increases by ∼30%–40% as the size increases in each SST experiment, and the increasing tendency of steady-state TC intensity with size is comparable under different SST experiments (see the linear regression line in Fig. 3a). Since both the inner-core and outer-core sizes can reflect the increasing tendency of TC steady-state intensity with size (Fig. 3), we will only focus on the inner-core size (RMW) to further investigate the effect of TC size on the TC energy cycle and steady-state intensity in the following discussion. Note that from Fig. 3a the largest steady-state RMW is ∼25 km, which seems to be smaller than those in some real TCs around their peak intensity (∼60 km; Li et al. 2022). This is understandable since in real atmosphere, many TCs cannot attain their idealized steady state and the intensification or RMW contraction processes could be interrupted by detrimental environmental dynamical processes (Emanuel 2000), resulting in the larger RMW around peak intensity than in idealized simulations.

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Steady-state maximum 10-m total wind speed V_m as a function of steady-state (a) RMW, (b) R17, and (c) R5. The dashed lines in (a) denote the linear regression line for SST28–SST31 with the corresponding slope (intercept) being 1.3, 1.0, 1.1, and 1.4 m s⁻¹ km⁻¹ (22.7, 32.7, 35.8, and 32.4 m s⁻¹), respectively.

Citation: Journal of the Atmospheric Sciences 82, 3; 10.1175/JAS-D-24-0042.1

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As mentioned above, Tao et al. (2020) and Rotunno (2022) attributed the increase in the maximum boundary layer tangential wind speed with size to the increase in supergradient wind in the boundary layer, which is confirmed in Fig. 4a. However, the intensity here in Fig. 3 is defined based on the wind near the surface (10 m), where there is often no supergradient wind (Li et al. 2020b). Figure 4b shows the difference between tangential wind speed and gradient wind speed at the lowest model level (25 m) at RMW. We can see that near the surface the wind is subgradient as in Li et al. (2020b). This means that the increase in maximum 10-m total wind speed with size cannot be explained by the supergradient wind in the boundary layer. Note that although the maximum boundary layer wind and maximum surface wind are inferred from different perspectives in theory, i.e., boundary layer dynamics and energetics, they are interrelated. With the increase in TC size, the increase in surface wind speed (Fig. 3) indicates a higher surface friction by definition, which could result in a larger subgradient force or subgradient winds in the inner core (Fig. 4b). The higher subgradient force indicates a higher inward force and thus inward jet, which could eventually generate a higher acceleration of tangential wind in the inner core and thus supergradient winds in the whole boundary layer (Fig. 4a; Kepert and Wang 2001; Li et al. 2020b). In the following discussion, we do not go into detail on the boundary layer dynamics and supergradient wind. Instead, an energetic analysis, which is directly related to the surface wind as mentioned earlier, will be discussed.

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Steady-state (a) maximum supergradient wind speed in the boundary layer and (b) difference between tangential wind speed and gradient wind speed at 25-m height at RMW as a function of steady-state RMW.

Citation: Journal of the Atmospheric Sciences 82, 3; 10.1175/JAS-D-24-0042.1

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4. Thermodynamic cycles of the TC heat engine

a. A revisit of TC heat engine’s theorem

Based on Emanuel (1986, 1991, 1997), the TC heat engine can be formulated from Bernoulli’s equation and the first law of thermodynamics. Along air parcel’s trajectories, Bernoulli’s equation including moist processes can be written (Li et al. 2023) as

$$d\left(\frac{1}{2}{\left|\mathbf{V}\right|}^2\right)+d(\text{Γ}z)+{\alpha }_{d}dp+\mathit{Fdl}+\text{Γ}{r}_{T}dz=0,$$(1)

where $\left|\mathbf{V}\right|$ is the total wind speed, Γ is the acceleration of gravity, z and p are the height and pressure, α_d is the specific volume of dry air, F is the viscous drag force per unit mass of dry air, l is the air parcel’s trajectories, and r_T is the mixing ratio of total water. The first law of thermodynamics including moist processes can be written (Pauluis and Zhang 2017) as

$$\mathit{Tds}=dh-{\alpha }_{d}dp-{\displaystyle \sum _{\mu =\upsilon ,l,i}{g}_{\mu }d{r}_{\mu }},$$(2)

where T is the air parcel temperature, s and h are the moist entropy and enthalpy per unit mass of dry air, g_υ, g_l, and g_i are the specific Gibbs free energy for water vapor, liquid water, and ice, and r_υ, r_l, and r_i are the mixing ratios of water vapor, liquid water, and ice. Following Pauluis and Zhang (2017), the entropy and Gibbs free energy are calculated with the thermodynamic reference state defined as liquid water at the freezing temperature, and the detailed definition can be found in the appendix of Pauluis and Zhang (2017) or Li et al. (2023).

Integrating Eqs. (1) and (2) over a closed thermodynamic cycle and combining them give

$$\underset{W_{\text{max}}}{\underbrace{{\displaystyle \oint \mathit{Tds}}}}=\underset{W_{\text{KE}}}{\underbrace{{\displaystyle \oint \mathit{Fdl}}}}+\underset{W_P}{\underbrace{{\displaystyle \oint \text{Γ}{r}_{T}dz}}}-\underset{G_P}{\underbrace{\oint \sum _{\mu =\upsilon ,l,i}{g}_{\mu }d{r}_{\mu }}},$$(3)

where W_KE can be defined by integrating Eq. (1) as

$$\underset{W_{\text{KE}}}{\underbrace{{\displaystyle \oint \mathit{Fdl}}}}=\underset{W_M}{\underbrace{-{\displaystyle \oint {\alpha }_{d}dp}}}-\underset{W_P}{\underbrace{{\displaystyle \oint \text{Γ}{r}_{T}dz}}}.$$(4)

Each term in Eqs. (3) and (4) denotes a work per unit mass of dry air circulating the thermodynamic cycle: The term W_max is the theoretical maximum work that would be done by the TC Carnot heat engine; W_M is the total mechanical work dissipation/production including the dry-air-associated mechanical work dissipation W_KE and water-substance-associated mechanical work dissipation W_P; and G_P is the Gibbs penalty. The terms G_P and W_P explain the effects of the hydrological cycle on the energy cycle, and both limit the generation of kinetic energy W_KE in the cycle (Pauluis and Zhang 2017).

It is our interest to understand how TC size affects those works over the thermodynamic cycle [i.e., Eqs. (3) and (4)] and TC steady-state intensity. One of the key issues in understanding the TC energy cycle is how to define the closed thermodynamic/energy cycle, as air parcel’s movement in TCs is often highly turbulent and not closed. For simplicity, Emanuel (1986, 1991, 1997) omitted the hydrological cycle in the TC heat engine and qualitatively divided the thermodynamic cycle of air parcels in Eulerian coordinates into four legs: spiraling toward the TC eyewall near the ocean surface from a periphery/ambient radius in leg 1, ascending in the eyewall and then flowing out to a large radius in leg 2, descending following the outflow downstream in leg 3, and further descending to the starting point of leg 1 in leg 4. Emanuel (1986, 1991, 1997) assumed that (i) heating and cooling Tds mainly occur near the ocean surface by enthalpy flux (leg 1) and the lower stratosphere by infrared radiation (leg 3), respectively, and (ii) viscous dissipation Fdl mainly occurs near the ocean surface by friction (leg 1). Emanuel (1997) further assumed that the balance between the production and dissipation of mechanical work of air parcels over the cycle is maintained by the local balance between the available production rate and dissipation rate of energy over the ocean surface near the RMW and derived a closed expression for the theoretical maximum intensity in terms of surface wind speed V_E−mpi:

$${\mathbf{V}}_{E-\text{mpi}}^2=\frac{C_k}{C_D}ϵT_s(s_0^{*}-s_b),$$(5)

where T_s is the SST, $s_0^{*}$ and s_b are the saturation entropy at the SST and entropy of the boundary layer air, and ϵ is the thermodynamic efficiency of the idealized Carnot cycle, defined as

$$ϵ=\frac{T_s-T_o}{T_s},$$(6)

where T_o is the mean air temperature of TC’s outflow in leg 3. Note that V_E−mpi [Eq. (5)] can be also derived based on a differential Carnot engine assumption (Emanuel 2018; Rousseau-Rizzi and Emanuel 2019).

Although Emanuel’s model simplifies the TC heat engine problem, the idealized four-leg movement in Eulerian coordinates could not be found (Fang et al. 2019; Li et al. 2023). In addition, the surface-wind-based theoretical maximum intensity [Eq. (5)] shares a similar mathematical equation to the gradient-wind-based theoretical maximum intensity (Emanuel 1986; Emanuel and Rotunno 2011; Rousseau-Rizzi and Emanuel 2019), which has been shown to fail to capture the increasing tendency of TC intensity with size (Rotunno 2022). Figure 5a shows the performance of the simplified theoretical maximum intensity V_E−mpi [Eq. (5)] for all simulations using the simulated steady-state results at the RMW following Rousseau-Rizzi and Emanuel (2019). The air–sea enthalpy disequilibrium $[T_s(s_0^{*}-s_b)]$ is directly diagnosed from the model output as in Li et al. (2020a). The outflow temperature T_o is defined as the temperature at the intersection of the zero-tangential-wind contour and the absolute angular momentum surface across the maximum wind speed as in Rousseau-Rizzi and Emanuel (2019) and Li et al. (2020a). As expected, the simplified theoretical intensity changes little (less than 10%) with the increase in size for each SST experiment, and thus, the significant increasing tendency of the simulated TC intensity with increasing size (Fig. 3) cannot be captured by the simplified model (Fig. 5a). This also indicates that at a large TC size in the current model configuration, the simulated intensity might surpass the simplified theoretical intensity, as confirmed by the additional simulations with a larger initial size (not shown), which occurs frequently in idealized simulations and often termed superintensity (Wang and Xu 2010; Rousseau-Rizzi and Emanuel 2019; Li et al. 2020a). Since C_k/C_D and T_s are constants during steady state for each SST experiment, the variation of the theoretical intensity depends on the idealized thermodynamic efficiency ϵ and the air–sea entropy disequilibrium $(s_0^{*}-s_b)$ [Eq. (5)]. The diagnosed ϵ and $(s_0^{*}-s_b)$ are shown in Figs. 5b and 5c, from which we can see that both increase significantly with increasing SST but change little with increasing size. Therefore, the simplified model [Eq. (5)] cannot provide insights into the physical understanding on the dependence of TC intensity on size.

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(a) Steady-state simplified theoretical maximum intensity V_E−mpi, (b) idealized thermodynamic efficiency of the Carnot cycle ϵ, and (c) air–sea entropy disequilibrium $(s_0^{*}-s_b)$ as a function of steady-state RMW.

Citation: Journal of the Atmospheric Sciences 82, 3; 10.1175/JAS-D-24-0042.1

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b. Reconstruction of the TC thermodynamic cycles

To define a closed TC energy/thermodynamic cycle and understand the effect of TC size on TC energy cycle and intensity, we performed an analysis based on a novel methodology named Mean Airflow as Lagrangian Dynamic Approximation (MAFALDA; Mrowiec et al. 2016; Pauluis and Zhang 2017) for the simulated steady-state results. The MAFALDA has been widely used in the literature (e.g., Mrowiec et al. 2016; Pauluis and Zhang 2017; Fang et al. 2019; Li et al. 2023). The first step of MAFALDA is to transform the steady-state net upward mass flux in Eulerian coordinates ρw′(z, r) into isentropic coordinates ρw′(z, θ_e), where w′ is the perturbation vertical velocity deviating from its domain-averaged value and θ_e is the equivalent potential temperature [see Pauluis and Zhang (2017) or Li et al. (2023) for the detailed calculation]. The data in Eulerian coordinates used for the coordinate transformation are within the domain of 1500-km radius, and our test indicates that the results below are qualitatively unchanged when the 3000-km radius is used. The TC’s isentropic streamfunction Ψ(z, θ_e) then can be computed as the isentropic integral of ρw′(z, θ_e). Since the isolines of isentropic streamfunction account for the mean trajectories of air parcels in isentropic coordinates (Pauluis and Mrowiec 2013), the TC thermodynamic cycles can be extracted along the isolines of isentropic streamfunction (hereafter MAFALDA trajectories), without considering the trajectory of any individual air parcel or turbulent flow.

To link the energy cycle and TC intensity, we only considered the MAFALDA trajectories across the maximum 10-m total wind speed. Figures 6a–d show the selected MAFALDA trajectories of all simulations, in which “A,” “B,” and “C” denote the points of the minimum θ_e, the maximum near-surface θ_e, and the highest height, respectively, for convenience. The legs of “AB,” “BC,” and “CA” generally represent the TC inflow leg, ascending leg, and descending leg, respectively, similar to the TC secondary circulation in Eulerian coordinates (Pauluis and Zhang 2017; Fang et al. 2019). As expected, the maximum 10-m total wind speed (see the black asterisk) occurs near the point of the maximum near-surface θ_e in the MAFALDA cycle. The thermodynamic variables can be interpolated along the MAFALDA trajectories to further understand the thermodynamic processes of the TC heat engine. Figures 6e–h show the T–s diagram along the MAFALDA trajectories based on the interpolated T and s. Note that as in Pauluis and Mrowiec (2013) and Pauluis and Zhang (2017), to filter out fast and reversible oscillatory motions, the interpolation for each variable is based on its mass-weighted conditional average. Morphologically, the MAFALDA trajectories in the z–θ_e diagram are simliar to those in the T–s diagram because temperature and entropy can be thought of as a proxy for height and equivalent potential temperature, respectively. We can see from Fig. 6 that the TC heat engine gains energy/entropy at warmer temperatures (leg AB) and loses energy/entropy at colder temperatures (legs BC and CA), and thus, the external heat energy can be continuously extracted and converted into mechanical energy by the thermodynamic cycle as a Carnot heat engine. With increasing SST, the θ_e or s of the whole MAFALDA trajectories increases and the ascending leg tends to attain a higher height (see point C), indicating a stronger secondary circulation.

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Steady-state MAFALDA trajectories across the location of the maximum 10-m total wind speed in (a)–(d) z–θ_e coordinates and (e)–(h) T–s coordinates. The letters A, B, and C mark the points of the minimum θ_e or s, the maximum near-surface θ_e or s, and the highest height or the lowest temperature, respectively. The letter O in (a)–(d) denotes the point around the minimum near-surface θ_e. The black asterisk “*” in (a)–(d) denotes the average location of the maximum 10-m total wind speed in the MAFALDA cycle for all simulations. The color bar indicates the initial RMW (km).

Citation: Journal of the Atmospheric Sciences 82, 3; 10.1175/JAS-D-24-0042.1

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A striking feature in Fig. 6 is that the minimum θ_e or s in the thermodynamic cycle (see point A) tends to decrease with increasing TC size, especially for the experiments with lower SSTs. This means that the source entropy of the TC inflow leg following the peripheral or outer-core descending flows (leg CA) would be lower with a larger TC size. To understand how TC size affects entropy following the outer-core descending flows, the entropy distribution in Eulerian coordinates for different size experiments has been compared in Fig. 7. There is a low-entropy layer around 2–6-km heights away from the TC inner core, which is a typical structure in the tropics. The outer-core descending flows across the low-entropy layer correspond to the descending leg in the MAFALDA cycle. We can see from Fig. 7 that the entropy in the low-entropy layer is redistributed/fragmented to a larger radially outward extent and the minimum entropy in the low-entropy layer is lower with a larger TC size for all SST experiments. Considering entropy redistribution is often associated with convective activities (Emanuel 1997; Pauluis and Zhang 2017), the vertical motion with 1 and −1 cm s⁻¹ contours has been shown in Fig. 7. It is clear that the upward and downward motions or convective activities in the outer core increase as TC size increases. This is understandable since a larger TC size indicates a broader overall wind profile (Fig. 2) and thus a higher overall surface enthalpy flux (Fig. 8) and a condition more favorable for convective activities (Xu and Wang 2010). With more or stronger downward motions, more dry air would be transported downward and stronger adiabatic subsidence warming and thus stronger longwave radiative cooling would be induced (Houze 2014). This means that both the drying and warming effects associated with downward motions could lower the local entropy. Note again that the sequence of events described here may not occur instantaneously but is more likely an adjustment process under the effect of TC size, which has been keeping the memory of the initial size throughout the development and steady-state stages. As shown in Fig. 7, the low (high) entropy patches over the TC outer-core region generally correspond with downward (upward) motions.

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Radial–vertical cross sections of the steady-state entropy s (shading; J kg⁻¹ K⁻¹) and vertical velocity at −1 cm s⁻¹ (black contour) and 1 cm s⁻¹ (white contour) for the simulations with the steady-state RMW (a)–(d) <15, (e)–(h) between 15 and 20, and (i)–(l) >20 km, respectively.

Citation: Journal of the Atmospheric Sciences 82, 3; 10.1175/JAS-D-24-0042.1

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Radial distribution of the steady-state surface enthalpy flux inside (left subpanel) and outside (right subpanel) the radius of 300 km with different colors showing the simulations with different initial RMWs (km).

Citation: Journal of the Atmospheric Sciences 82, 3; 10.1175/JAS-D-24-0042.1

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To confirm the contribution by outer-core downward motions to the decrease in entropy, an entropy budget analysis has been performed for the period 24 h before the steady state is reached. Considering that the tendency terms of potential temperature θ and water vapor mixing ratio r_υ can be directly output from CM1, for simplicity, we use the entropy budget equation in terms of θ and r_υ as in Rotunno and Emanuel (1987) and Rousseau-Rizzi et al. (2021), i.e., ∂s/∂t = (c_p/θ)(∂θ/∂t) + (L_υ/T)(∂r_υ/∂t), where c_p and L_υ are the heat capacity of dry air and constant latent heat of vaporization, respectively. Because the qualitative results for all simulations are similar, only results for simulations in SST28 with the steady-state RMW > 20 km are shown in Fig. 9 as a representative example. We can see that the outer-core downward motions and low entropy patches (contours) are consistent well with the negative tendency of entropy based on all budget terms in CM1 (Figs. 9a,b). Figures 9c–e show the budget terms associated with advection, radiation, and turbulence, which dominate the entropy tendency over the TC outer-core region, i.e.,

$$\frac{\partial s}{\partial t}=\underset{\text{Advection}}{\underbrace{-\frac{c_p}{\theta }\left(u\frac{\partial \theta }{\partial r}+w\frac{\partial \theta }{\partial z}\right)-\frac{L_{\upsilon }}{T}\left(u\frac{\partial r_{\upsilon }}{\partial r}+w\frac{\partial r_{\upsilon }}{\partial z}\right)}}+\underset{\text{Radiation}}{\underbrace{\frac{c_p}{\theta }{\dot{\theta }}_{\text{ra}}}}+\underset{\text{Turbulence}}{\underbrace{\frac{c_p}{\theta }{\dot{\theta }}_{\text{tur}}+\frac{L_{\upsilon }}{T}{\dot{r}}_{\upsilon ,\text{tur}}}},$$(7)

where u and w are radial and vertical velocities, ${\dot{\theta }}_{\text{ra}}$ and ${\dot{\theta }}_{\text{tur}}$ are the tendencies of θ due to radiation and turbulence, and ${\dot{r}}_{\upsilon ,\text{tur}}$ is the tendency of r_υ due to turbulence. As expected, both the advection and radiation terms contribute to the decrease in entropy in the low entropy patches (Figs. 9c,e), and the negative advection is mainly due to the negative vertical advection of r_υ (Fig. 9f). Therefore, from a Lagrangian perspective, it is the stronger longwave radiative cooling associated with downward motions that result in the lower minimum θ_e or s of air parcel in the descending flows with a larger TC size (Fig. 9e). In addition, the turbulence term tends to induce a weak decrease (increase) in entropy in the upper (lower) part of the low entropy patches due to the vertical gradient of entropy (Fig. 9d).

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Radial–vertical cross sections of entropy s at 190 and 200 J kg⁻¹ K⁻¹ (black contours; as a reference for the low entropy patches) and (a) vertical velocity (shading; cm s⁻¹) or entropy tendency (shading; J kg⁻¹ K⁻¹ h⁻¹) based on (b) all budget terms in CM1, (c) advection terms, (d) turbulence term, (e) radiation term, and (f) vertical advection of r_υ averaged over the 24-h period before the steady state is reached for simulations in SST28 with the steady-state RMW > 20 km.

Citation: Journal of the Atmospheric Sciences 82, 3; 10.1175/JAS-D-24-0042.1

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The above analysis confirms that downward motions tend to lower entropy. As a result, the minimum θ_e or s following the descending leg in the MAFALDA cycle would be lower with increasing TC size and outer-core convective activities (Fig. 6). Note that with more convective activities, the more or stronger outer-core upward motions would also induce increases in entropy, as depicted by those high-entropy patches (Figs. 7 and 9). However, those outer-core upward motions along the high-entropy patches correspond to the shallow thermodynamic cycles, which are less pertinent to TC intensity (Pauluis and Zhang 2017; Fang et al. 2019). We thus do not go into its details. Note also that although with increasing SST, the outer-core convective activities increase (Fig. 7) and the minimum θ_e or s of the MAFALDA cycle does not decrease but increases (Fig. 6). This is because the whole troposphere is warmed by the ocean surface and thus would be warmer with a higher SST (Fig. 7).

Another striking feature in Fig. 6 is that although the TCs with a larger size show a lower source entropy of the inflow leg as mentioned above, they attain almost the same entropy at the end of the inflow leg (see point B in Fig. 6). This means that the entropy change along the inflow leg is larger with a larger size. Namely, the TC heat engine with a larger size obtains more entropy in the inflow leg. This is understandable since the entropy change or entropy supply is mainly determined by the surface enthalpy flux along the inflow leg as mentioned earlier (e.g., Fang et al. 2019), which mainly depends on the near-surface wind speed (not shown) and is thus higher in the larger TCs (Fig. 8). The analyses above (Figs. 6–9) indicate that the main effect of TC size on the TC heat engine is twofold. The first is to modulate the entropy around the end of the descending leg or the start of the inflow leg in the TC thermodynamic cycle, by affecting the convective activities over the TC peripheral or outer-core region. The second is to modulate the energy supply of the inflow leg in the TC heat engine. Both effects are due to the modulation of the surface enthalpy flux by TC size. In addition, as seen from Fig. 6, both effects are weaker for a higher SST. This is probably because with increasing SST, the overall entropy flux would be enhanced (Fig. 8), and thus the modulation of entropy flux by size becomes less significant. The effect of SST on the size dependences of TC thermodynamic cycle and TC intensity will be discussed in more detail later.

Since in the TC heat engine the external energy is mainly absorbed by the inflow leg (Fig. 6), a stronger energy supply or larger entropy change along the inflow leg indicates a higher energy input ${\int }_A^B\mathit{Tds}$ and thus a higher generation of kinetic energy by definition. Therefore, it is the second effect of TC size mentioned above, i.e., on the energy supply of the inflow leg, that directly affects the generation of kinetic energy in the TC heat engine. Note that although the source entropy of the inflow leg, i.e., the first effect, may affect the energy absorption of air parcel along the inflow leg, a lower source entropy does not necessarily mean a larger entropy increase or a higher generation of kinetic energy. As shown in Rousseau-Rizzi et al. (2021), a lower source entropy over the subsidence region may be associated with a decay of TC intensity on time scales of 10 or more days if without sufficient energy supply for the TC thermodynamic cycle. Note that the time for a parcel to travel along the inflow leg is often shorter than 2 days (Rotunno and Emanuel 1987; Rousseau-Rizzi et al. 2021). This means that the effect of TC size on the energy supply or entropy change along the inflow leg can be well reflected in the TC heat engine and TC steady-state intensity for all simulations (cf. Fig. 1).

The link between TC intensity and difference between the TC inner-core entropy and outer-core entropy has been discussed in many previous studies (e.g., Emanuel 1986; Gu et al. 2015). Figure 10 shows the difference between the entropy at the maximum 10-m total wind speed and the minimum entropy over the TC outer-core region for all simulations. Note that we focus on the minimum entropy over the TC outer-core region because those descending flows that would be followed by inflow flows in the energy cycle are often coupled with the low entropy anomaly as mentioned above (cf. Fig. 7). As expected, although the minimum entropy over the TC outer-core region shows a clear decreasing tendency with TC size (Fig. 10a), there is no significant trend in the inner-core entropy (Fig. 10b). This is consistent with the increasing difference between the inner-core entropy and outer-core entropy as the TC size increases (Fig. 10c). Based on the results here from Fig. 10 together with that from Figs. 6–9, we hypothesize that the dependence of the steady-state TC intensity on TC size in the simulations is related to the modulation of the entropy change along the TC inflow leg in the TC heat engine by TC size. This hypothesis will be further evaluated based on detailed energy budgets in the next subsection. Note that the effects of TC size on outer-core minimum entropy and difference between inner-core entropy and outer-core entropy decrease with increasing SST (Fig. 10) as mentioned before (Fig. 6).

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Steady-state (a) minimum entropy outside 100-km radius, (b) entropy at the location of the maximum intensity, and (c) difference between the entropy at the location of the maximum intensity and the minimum entropy outside 100-km radius as a function of the steady-state RMW.

Citation: Journal of the Atmospheric Sciences 82, 3; 10.1175/JAS-D-24-0042.1

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c. Energy budgets of the TC heat engine

Based on the closed energy cycle (Fig. 6), the energy conversion of the TC heat engine can be diagnosed based on Eqs. (3) and (4) with those thermodynamic variables interpolated along the MAFALDA trajectories using the simulated steady-state results. Figures 11a–e show those diagnosed energy terms in Eq. (3), with the dry-air-associated mechanical dissipation, i.e., the generation of kinetic energy W_KE, calculated using Eq. (4). We can see that the residual term of Eq. (3) (Fig. 11e) is much smaller than the other terms in all simulations, indicating that the energy budget based on MAFALDA can well capture the energy conversion in the simulated TCs. This also indicates that the assumption of quasi-steady thermodynamic flow in the MAFALDA framework is well satisfied during the defined steady state. As expected, the generation of kinetic energy W_KE increases significantly (more than 40%) as TC size increases in each SST experiment (Fig. 11b), which is mainly due to the increase in the theoretical maximum work W_max (Fig. 11a). Note that although the works associated with the hydrological cycle W_P and G_P limit the generation of kinetic energy (Pauluis and Zhang 2017), they show a rather smaller variation with TC size compared with W_KE and thus cannot explain the increasing tendency of W_KE with TC size (Figs. 11c,d). This result is consistent with that of Wang and Lin (2021), whose budgets show that the degree of irreversibility associated with the hydrological cycle does not depend on the wind structure of a TC.

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Steady-state (a) theoretical maximum work W_max ($\oint \mathit{Tds}$), (b) dry-air-associated mechanical dissipation W_KE ($-\oint {\alpha }_{d}dp-W_P$), (c) water-substance-associated mechanical work W_P ($\oint \text{Γ}{r}_{T}dz$), (d) Gibbs penalty G_P ($-\oint {\displaystyle {\sum }_{\mu =\upsilon ,l,i}{g}_{\mu }d{r}_{\mu }}$), and (e) the residual term W_max − W_KE − W_P − G_P of Eq. (3) based on the MAFALDA cycle as a function of steady-state RMW. (f) Square of the maximum intensity $V_m^2$ as a function of W_KE.

Citation: Journal of the Atmospheric Sciences 82, 3; 10.1175/JAS-D-24-0042.1

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The energy budget above shows that the size dependence of the steady-state TC intensity in simulations corresponds well with the size dependence of W_KE ($\oint \mathit{Fdl}$). This is because most of the kinetic energy generated in the TC heat engine is dissipated by the ocean surface friction, which depends on the surface wind speed and thus the TC intensity. If we assume that the dissipation mainly occurs near the ocean surface (leg OB in Fig. 6) and use the slab boundary layer–based drag force formula $F\approx C_D{\left|{\mathbf{V}}_s\right|}^2/H$ (e.g., Wang et al. 2021) with $\left|{\mathbf{V}}_s\right|=\left|{\mathbf{V}}_m\right|\tilde{V}$, we can rewrite $\oint \mathit{Fdl}$ as

$${\displaystyle \oint \mathit{Fdl}}\approx {\mathbf{V}}_m^2{\displaystyle {\int }_O^B\frac{C_D}{H}{\tilde{V}}^{2}dl},$$(8)

where V_s is the 10-m total wind speed, V_m is the maximum 10-m total wind speed, i.e., TC intensity, $\tilde{V}$ is the nondimensional 10-m total wind speed, and H is the depth of the slab boundary layer. Considering both the points B and O change little with the increase in TC size (Fig. 6) and the nondimensional near-surface wind profile $\tilde{V}$ from RMW to the outer core keeps a similar pattern with different TC sizes (Fig. 12), we simply assume that the nondimensional wind profile $\tilde{V}$ along the near-surface trajectory l changes little with TC size. Therefore, we may expect a linear relationship between W_KE and $V_m^2$ based on Eq. (8), which is verified in Fig. 11f. This means that TC size modulates the maximum intensity mainly by affecting the generation of kinetic energy W_KE in the TC heat engine. Note that as shown in Eq. (8), an increase in W_KE does not just mean an increase in the maximum 10-m total wind speed, i.e., the terminus of the inflow leg (Fig. 6), but an increase in the overall wind speed $\left|{\mathbf{V}}_m\right|\tilde{V}$ along the near-surface inflow leg.

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(a) Distributions of the normalized steady-state 10-m total wind speed along the normalized radius for different SSTs with different colors showing the simulations with different initial RMWs (km; color bar).

Citation: Journal of the Atmospheric Sciences 82, 3; 10.1175/JAS-D-24-0042.1

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It is still unclear whether the theoretical maximum work W_max ($\oint \mathit{Tds}$) that constrains W_KE is largely affected by the entropy change along the inflow leg (Fig. 6). In addition, another remaining issue is why the change rate of the steady-state intensity or W_KE with TC size is similar in different SST experiments (Figs. 3a and 11b) even though the increasing tendency of the entropy change along the TC inflow leg with TC size is smaller in the experiment with higher SST as mentioned above (Figs. 6 and 10). To address the two issues, a further analysis for W_max is performed. Theoretically, in a Carnot heat engine, W_max is determined by the total energy source Q_in and Carnot efficiency ϵ_C, namely, W_max = ϵ_CQ_in. Along the MAFALDA cycle, the external heating increment dq, Q_in, and total energy sink Q_out are defined (Pauluis and Zhang 2017) as

$$dq=\mathit{Tds}+{\displaystyle \sum _{\mu =\upsilon ,l,i}{g}_{\mu }d{r}_{\mu }},$$(9a)

$$Q_{\text{in}}={\displaystyle \oint \text{max}}(dq,0),$$(9b)

and

$$Q_{\text{out}}={\displaystyle \oint \text{min}}(dq,0).$$(9c)

With Q_in and Q_out, the average temperatures of total energy source T_in and sink T_out over the MAFALDA cycle can be calculated by

$$\frac{Q_{\text{in}}}{T_{\text{in}}}={\displaystyle \oint \max }\left(\frac{dq}{T},0\right),$$(10a)

and

$$\frac{Q_{\text{out}}}{T_{\text{out}}}={\displaystyle \oint \min }\left(\frac{dq}{T},0\right).$$(10b)

The Carnot efficiency ϵ_C and the actual efficiency of TC heat engine ϵ_T then can be given by

$${ϵ}_C=\frac{T_{\text{in}}-T_{\text{out}}}{T_{\text{in}}},$$(11a)

and

$${ϵ}_T=\frac{W_{\text{KE}}}{Q_{\text{in}}}.$$(11b)

Considering the fact that the external heating by ${\displaystyle {\sum }_{\mu =\upsilon ,l,i}{g}_{\mu }d{r}_{\mu }}$ is small compared with that by Tds in the MAFALDA cycle (Fig. 11) and the entropy input mainly occurs in the inflow leg (leg AB in Fig. 6), it is not unrealistic to assume that Q_in is dominated by ${\int }_A^B\mathit{Tds}$ and thus W_max can be simplified as

$${\displaystyle \oint \mathit{Tds}}\approx {ϵ}_C{\displaystyle {\int }_A^B\mathit{Tds}}.$$(12)

Figure 13a shows the simplified W_max calculated by Eq. (12). Although the simplified W_max shows a lower value than the actual W_max in each simulation (Fig. 11a), it well captures the increasing tendency of W_max with TC size. Figures 13b and 13c show the simplified total energy source ${\int }_A^B\mathit{Tds}$ and the average temperature of total energy source T_in, which is dominated by the temperature in the inflow leg AB. We can see that ${\int }_A^B\mathit{Tds}$ significantly increases but T_in changes little with TC size, indicating that the increasing tendency of ${\int }_A^B\mathit{Tds}$ with TC size (Fig. 13b) is mainly due to the increasing entropy change along the inflow leg ${\int }_A^{B}ds$ with TC size (Fig. 6). Figure 13d shows the average temperature of total energy sink T_out. As in T_in, T_out also does not show any significant variation with TC size. As a result, the Carnot efficiency ϵ_C and thus the actual efficiency of TC heat engine ϵ_T [≈ϵ_C − (W_P + G_P)/Q_in] change little with TC size (Figs. 13e,f). These results confirm that the increasing tendency of W_max ($\approx {ϵ}_C{\int }_A^B\mathit{Tds}$) and thus W_KE ($\approx {ϵ}_T{\int }_A^B\mathit{Tds}$) with TC size is mainly due to the increasing entropy change along the inflow leg ${\int }_A^{B}ds$ with TC size.

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Steady-state (a) theoretical maximum work W_max diagnosed by ${ϵ}_C{\int }_A^B\mathit{Tds}$, (b) energy input ${\int }_A^B\mathit{Tds}$, (c) average temperatures of total energy source T_in, (d) average temperatures of total energy sink T_out, (e) Carnot efficiency ϵ_C, and (f) actual efficiency of the TC heat engine ϵ_T as a function of the steady-state RMW.

Citation: Journal of the Atmospheric Sciences 82, 3; 10.1175/JAS-D-24-0042.1

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In addition, we can see from Fig. 13d that T_out decreases with increasing SST, which is associated with the higher ascending leg and thus higher outflow layer in the experiment with higher SST as mentioned above (Fig. 6). The lower T_out results in the higher ϵ_T and thus ϵ_C in the experiment with a higher SST (Figs. 13e,f). Therefore, although the increasing tendency of the entropy change along the TC inflow leg and thus ${\int }_A^B\mathit{Tds}$ with TC size is smaller in the experiment with a higher SST (Figs. 6, 10, and 13b), the higher ϵ_T and ϵ_C with a higher SST can amplify the increasing tendency of W_max ($\approx {ϵ}_C{\int }_A^B\mathit{Tds}$) and W_KE ($\approx {ϵ}_T{\int }_A^B\mathit{Tds}$) with TC size. As a result, the increasing tendency of W_max and W_KE and thus steady-state TC intensity with TC size are comparable under different SST experiments (Figs. 3, 11a, 11b, and 13a).

5. Conclusions and discussion

Previous numerical studies have found that TCs with a larger initial size tend to obtain a larger steady-state size with a higher steady-state intensity (e.g., Xu and Wang 2010, 2018; Kilroy and Smith 2017; Tao et al. 2020; Rotunno 2022; Fei and Wang 2024; Li et al. 2024). The positive correlation between the initial and steady-state sizes reflects a strong memory of the vortex structure (Xu and Wang 2010; Tao et al. 2020; Rotunno 2022), and the higher steady-state intensity with a larger initial or steady-state size has been attributed to a larger contribution of the supergradient wind to TC intensity based on the TC boundary layer dynamics (Tao et al. 2020; Rotunno 2022). This study revisits the effect of TC size on TC steady-state intensity based on the energetics using the isentropic analysis without considering the details of supergradient wind or boundary layer dynamics. The data used in this study are from a series of axisymmetric numerical simulations by varying the initial size of the TC vortex.

Results show that TC size affects the TC energy cycle and the steady-state intensity mainly by affecting the surface enthalpy flux and thus the energy supply to the TC heat engine. With a larger TC size, the overall surface enthalpy flux is higher and thus the inflow leg of the TC heat engine would absorb more entropy or energy from the ocean surface. A higher entropy increase along the inflow leg results in a higher theoretical maximum work and thus higher generation of kinetic energy in the TC heat engine. The higher generation of kinetic energy explains why the TC with a larger size has a higher steady-state intensity. It is also found that with increasing SST and thus surface enthalpy flux, the effects of TC size on surface enthalpy flux would become less important. The increasing rate of the entropy change along the inflow leg in the TC heat engine with TC size is thus smaller in the experiment with a higher SST. However, a higher SST would result in a stronger secondary circulation and a lower temperature of the energy sink and thus a higher Carnot efficiency. The higher Carnot efficiency can amplify the increasing tendency of the theoretical maximum work and generation of kinetic energy with TC size. As a result, the change rates of generation of kinetic energy and thus the steady-state TC intensity with TC size are similar under different SST experiments.

We should point out that in this study we focus on the effect of the overall (both inner core and outer core) size rather than the RMW on steady-state intensity, although we used the RMW as a proxy for the overall size for convenience. Actually, an increase in steady-state RMW with outer-core size unchanged is often combined with a decrease in steady-state intensity based on the theory of Emanuel and Rotunno (2011). Rotunno and Bryan (2012) showed that with a higher horizontal mixing length, the steady-state RMW tends to be larger and the steady-state intensity tends to be lower, as shown in Figs. 14a and 14b. Tao et al. (2020) and Rotunno (2022) also showed that the dependence of steady-state intensity on the RMW can be modified by the horizontal mixing length. However, the change in wind profile with horizontal mixing length is only in the inner core. The outer-core size or wind profile is almost unchanged with horizontal mixing length (Figs. 14c,d). This means that the effect of horizontal mixing length on the steady-state intensity is more likely a local effect and is different with the effect of overall size. Note also that in this study we only focus on the TC steady state. As mentioned earlier, in idealized simulations TCs after their steady state often gradually evolve toward an equilibrium state, during which the memory of initial size will gradually disappear and the final size is mainly determined by the boundary conditions and system properties of the model (Frisius 2015), as shown in Figs. 15b–d. As a result, the intensity in the equilibrium state is also independent of the initial size, as shown in Fig. 15a. We notice that in some idealized simulations TCs after their steady state tend to decay considerably (Rousseau-Rizzi et al. 2021), which does not occur in our simulations (Fig. 15). Our preliminary tests found that this difference could be caused by different model setups, such as the cloud microphysical scheme. However, a detailed analysis is beyond the scope of this study.

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Time series of (a) the maximum 10-m wind speed, (b) the RMW, and (c) the radius of 17 m s⁻¹ wind before and during the defined steady state. (d) Radial distributions of the steady-state 10-m total wind speed. The experiment is as in SST29 but with different colors showing the simulations with different horizontal mixing lengths (m).

Citation: Journal of the Atmospheric Sciences 82, 3; 10.1175/JAS-D-24-0042.1

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Time series of (a) the maximum 10-m wind speed, (b) the RMW, (c) the radius of 17 m s⁻¹ wind, and (d) the radius of 5 m s⁻¹ wind from the experiment as SST29 but only for the simulations with the initial RMW being 40, 80, and 120 km for convenience. All simulations are integrated for 100 days with 3-h output.

Citation: Journal of the Atmospheric Sciences 82, 3; 10.1175/JAS-D-24-0042.1

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This study suggests that the increase in overall sea surface enthalpy flux under a TC can enhance the steady-state TC intensity by increasing the energy input into the TC heat engine. Some previous studies through sensitivity simulations found that the sea surface enthalpy flux immediately outside the eyewall may be detrimental to TC steady-state intensity (Xu and Wang 2010) by facilitating outer rainbands, which tend to reduce the inflow toward the eyewall and thus the eyewall updraft (Wang 2009). This means that there are some competing processes involved regarding the effects of the sea surface enthalpy flux on TC intensity, which will be a good topic for future exploration. In addition, we only focus on the effect of the initial size on steady-state intensity in a quiescent idealized environment. In addition to the initial size, the steady-state size and intensity may be affected by many other factors, such as the horizontal mixing length (Rotunno and Bryan 2012), as mentioned above, and various environmental factors (Emanuel 1986). Future efforts should be given to understand the relationship between size and intensity using more realistic simulations or observational data. Finally, in this study, we only provide a qualitative explanation for the dependence of the steady-state TC intensity on TC size. It will be a good topic for a future study to quantify the dependence of TC steady-state intensity on TC size in theory.

Data availability statement.

The CM1 model was summarized in Bryan and Fritsch (2002), and the source code was downloaded from https://www2.mmm.ucar.edu/people/bryan/cm1/. Initial conditions and model configuration files are available at https://box.nju.edu.cn/d/1960326ce70945e7b962/.