1. Introduction

Upon the genesis of a tropical depression under favorable environmental conditions, such as high sea surface temperature (SST) and high heat content in the uppermost ocean layer, and weak vertical wind shear (Gray 1979), the cyclonic low pressure system may develop into a tropical cyclone (TC) of tropical storm strength with a maximum sustained surface wind speed higher than 17 m s⁻¹. In the absence of deleterious environmental effects, a TC often experiences a phase of rapid intensification (RI) associated with an increase in near-surface maximum wind speed of more than 15 m s⁻¹ day⁻¹, before it reaches its maximum intensity (Wang and Wu 2004). Over the past five decades or so, many efforts have been devoted to conceptualizing the physical mechanisms that are responsible for TC intensification. Montgomery and Smith (2014) summarized and compared four most prominent paradigms of TC intensification in the literature. These are the conditional instability of the second kind (CISK) paradigm (Charney and Eliassen 1964); the cooperative intensification paradigm (Ooyama 1964, 1969, 1982); the thermodynamic air–sea interaction intensification paradigm [viz., the wind-induced heat exchange feedback (WISHE); Rotunno and Emanuel 1987; Emanuel 1988, 1989, 1995, 1997; Emanuel et al. 1994]; and the rotating convective paradigm (Nguyen et al. 2008; Montgomery et al. 2009; Smith et al. 2009). Each of these paradigms give, to some extent, a qualitative explanation for the intensification processes,¹ but none of these concepts provides a quantitative measure of the intensification rate (IR) of a TC.

Unlike the maximum potential intensity (MPI), i.e., the maximum intensity a TC may reach during its lifetime under given favorable environmental conditions, the TC IR cannot be treated under the assumption of a system in balanced steady state. Moreover, the IR of a TC may be affected by many more factors than MPI, such as instantaneous storm intensity, the radial location of diabatic heating, and both the inner- and outer-core structures of the TC, etc. (Schubert and Hack 1982; Vigh and Schubert 2009; Pendergrass and Willoughby 2009; Rogers et al. 2013; Carrasco et al. 2014; Xu and Wang 2015, 2018a,b). Recent studies based on best track TC data have provided evidence of some robust relationships between the TC IR and the inner- and outer-core sizes (Carrasco et al. 2014; Xu and Wang 2015, 2018a) and the TC intensity (Kaplan et al. 2010; Xu and Wang 2015, 2018a). Figure 1, which is adopted from Xu and Wang (2018a, see their Fig. 6), shows the frequency of subsequent 24-h IR against TC intensity, together with the 95th percentile of IR in different SST bins over the western North Pacific (WNP) during 1982–2015 based on the best track data from the Joint Typhoon Warning Center (JTWC). Overall, it can be seen that the IR increases with TC intensity if the 10-m sustained maximum wind speed (V_max) is below about 60–70 kt (or 30–36 m s⁻¹), an intensity range corresponding to TCs of about category-1 hurricane strength as defined by the Saffir–Simpson wind scale. In addition, it is obvious that the overall IR increases with SST, consistent with recent axisymmetric numerical simulations (Li et al. 2020).

Frequency of the subsequent 24-h intensification rate (IR) against instantaneous storm intensity (V_max in terms of maximum 1-min mean sustained 10-m wind speed) with the black, red, and green curves showing the spline fitted curves for the 95th percentile of IR for three 1.5°C intervals of SST starting at 27°, 28°, and 29°C, respectively [reproduced from Fig. 6 of Xu and Wang (2018a)].

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0393.1

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Frequency of the subsequent 24-h intensification rate (IR) against instantaneous storm intensity (V_max in terms of maximum 1-min mean sustained 10-m wind speed) with the black, red, and green curves showing the spline fitted curves for the 95th percentile of IR for three 1.5°C intervals of SST starting at 27°, 28°, and 29°C, respectively [reproduced from Fig. 6 of Xu and Wang (2018a)].

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0393.1

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Frequency of the subsequent 24-h intensification rate (IR) against instantaneous storm intensity (V_max in terms of maximum 1-min mean sustained 10-m wind speed) with the black, red, and green curves showing the spline fitted curves for the 95th percentile of IR for three 1.5°C intervals of SST starting at 27°, 28°, and 29°C, respectively [reproduced from Fig. 6 of Xu and Wang (2018a)].

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0393.1

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The observed initial increase in the TC IR with TC intensity is often explained by the balance dynamics extensively studied by Shapiro and Willoughby (1982), Schubert and Hack (1982), and Pendergrass and Willoughby (2009). These studies showed that storms at greater intensity tend to have a higher inner-core inertial stability and thus higher heating efficiency to intensify the TC. However, the IR in observations tends to decrease with TC intensity once the storm becomes stronger than 60–70 kt (cf. Fig. 1). To explain this, it has been suggested that in an intensifying TC getting close to its MPI, the enhanced heating efficiency is mostly offset by the surface frictional dissipation effect (Kaplan et al. 2010; Xu and Wang 2015). Kaplan et al. (2010) also hypothesized that TCs of a certain intensity, which are sufficiently well organized but relatively far from their MPI, are likely to undergo RI.

Although several previous studies have derived time-dependent equations for TC intensity change (viz., IR) under various assumptions based on concepts of different complexities (DeMaria 2009; Emanuel 1997, 2012; Ozawa and Shimokawa 2015; Emanuel and Zhang 2017), the intensity-dependent IR with the maximum at an intermediate intensity as shown in observations has not been well discussed yet. In this study, we augment the energetically based dynamical system model conceptualized by Ozawa and Shimokawa (2015) to account for the intensity dependence of the TC IR as evident from best track TC data analyses as presented in Kaplan et al. (2010) and Xu and Wang (2015, 2018a). Modifications to the simple energetically based dynamical system model of a TC based on the viewpoint of a non-steady-state Carnot heat engine is introduced in section 2. The modified model may serve to physically explain and quantify the observed intensity dependence of the TC IR. Some assumptions made in the derivation of the simplified model are evaluated in section 3 based on results from idealized axisymmetric numerical simulations. Major conclusions are drawn in the section 4.

2. A modified energetically based dynamical system model for TC intensification

In this section, we discuss a modification to an energetically based conceptual model, which allows us to capture and quantify the intensity dependence of the TC IR as obtained from the best track TC data over the WNP (Fig. 1). As in earlier studies, we start with the basic concept on energy production and frictional dissipation processes in a TC system, where the TC is energetically considered as a Carnot heat engine after it has developed a well-defined eyewall structure (cf. Kleinschmidt 1951; Emanuel 1988, 1997; Wang and Xu 2010; Schönemann and Frisius 2012). For a detailed elaboration of the thermodynamic fundamentals underlying this concept, we refer the readers to the paper by Ozawa and Shimokawa (2015). Our working hypothesis is that the TC intensifies, as measured by the increase in mechanical energy in the inner core, when there is a positive net energy gain rate due to the imbalance between the energy production and frictional dissipation rates. Here, we extend the commonly used framework to describe TC energetics by introducing a normalized inertial stability factor, which acts on the effective power generation, thereby modifying the intensification potential (IP) of the TC. The inertial stability factor is introduced to account for the dynamical efficiency of eyewall heating, thus making the conceptual model applicable to weak TC systems as will be detailed below.

Figure 2 gives three examples of the IP [given by $\left(C_D/H\right)E{V}_{\text{mpi}}^2$] with different power constants (n = 0.0, 0.5, and 1.0) in the parameterized dynamical efficiency, the frictional weakening rate [FR, given by $\left(C_D/H\right)\hspace{0.17em}{V}_{\max }^2$], and the resulting IR as a function of the instantaneous storm intensity (V_max) for V_mpi = 60 m s⁻¹ at a radius of r_mpi = 15 km. The MPI value chosen here is roughly corresponding to conditions with an SST of 29°C over the tropical WNP, and E is calculated according to Eq. (9). Other parameters used in the calculations are C_D = 2.4 × 10⁻³, C_k = 1.2 × 10⁻³, f = 5 × 10⁻⁵, and H = 3 km (as in Emanuel and Zhang 2017). Note that, for simplicity, we have not considered the contraction of r_m during the intensification stage of the TC and assumed r_m is the same as r_mpi in the examples shown. Considering r_m is usually larger than r_mpi for weak systems (Li et al. 2019), the IR on the weak V_max side would become smaller for n > 0. In case n = 0 (black curves in Fig. 2), which is equivalent to E = 1 (as in Emanuel 2012), the potential energy available can be fully used to intensify the TC and thus the IP for E = 1 (dotted black curve) is independent of the instantaneous TC intensity. Since in all cases, the frictional weakening rate FR (solid blue curve) increases with the square of V_max, the IR for E = 1 (dashed black curve) would be maximal when V_max = 0 and decreasing monotonously as the TC intensifies. In that case, a maximum IR of ~250 m s⁻¹ day⁻¹ is obtained, which has never been reported from nature. Although the maximum IR may be reduced with larger H, it still occurs at V_max = 0, which is not consistent with the finite-amplitude nature of TC genesis/development described in Emanuel (1989). Furthermore, the monotonous decrease in IR with increasing TC intensity is in contrast to the observed intensity dependence of the TC IR which exhibits a maximum at intermediate intensity (cf. Fig. 1).

Dependence of the TC intensification rate (IR, m s⁻¹ day⁻¹, dashed curves, left Y axis) on the TC intensity V_max, with V_mpi = 60 m s⁻¹, and r_mpi = 15 km for three cases, which differ in the dependence of intensification potential (IP) on dynamical efficiency E with n = 0.0 (black curves), 0.5 (red curves), and 1.0 (green curves). The corresponding IPs (m s⁻¹ day⁻¹, dotted curves, right Y axis) and the frictional weakening rate (FR, m s⁻¹ day⁻¹, solid blue curve, right Y axis) are also shown. The V_mpi of 60 m s⁻¹ is obtained for T_S = 29°C, C_D = 2.4 × 10⁻³, C_k=1.2 × 10⁻³, and a sounding with the convective available potential energy of about 1900 J. For simplification, the RMW (r_m) of the intensifying storm in the dynamical efficiency calculation [cf. Eq. (9)] is assumed tos be constant and equal to r_mpi. See text for further references.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0393.1

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Dependence of the TC intensification rate (IR, m s⁻¹ day⁻¹, dashed curves, left Y axis) on the TC intensity V_max, with V_mpi = 60 m s⁻¹, and r_mpi = 15 km for three cases, which differ in the dependence of intensification potential (IP) on dynamical efficiency E with n = 0.0 (black curves), 0.5 (red curves), and 1.0 (green curves). The corresponding IPs (m s⁻¹ day⁻¹, dotted curves, right Y axis) and the frictional weakening rate (FR, m s⁻¹ day⁻¹, solid blue curve, right Y axis) are also shown. The V_mpi of 60 m s⁻¹ is obtained for T_S = 29°C, C_D = 2.4 × 10⁻³, C_k=1.2 × 10⁻³, and a sounding with the convective available potential energy of about 1900 J. For simplification, the RMW (r_m) of the intensifying storm in the dynamical efficiency calculation [cf. Eq. (9)] is assumed tos be constant and equal to r_mpi. See text for further references.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0393.1

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Dependence of the TC intensification rate (IR, m s⁻¹ day⁻¹, dashed curves, left Y axis) on the TC intensity V_max, with V_mpi = 60 m s⁻¹, and r_mpi = 15 km for three cases, which differ in the dependence of intensification potential (IP) on dynamical efficiency E with n = 0.0 (black curves), 0.5 (red curves), and 1.0 (green curves). The corresponding IPs (m s⁻¹ day⁻¹, dotted curves, right Y axis) and the frictional weakening rate (FR, m s⁻¹ day⁻¹, solid blue curve, right Y axis) are also shown. The V_mpi of 60 m s⁻¹ is obtained for T_S = 29°C, C_D = 2.4 × 10⁻³, C_k=1.2 × 10⁻³, and a sounding with the convective available potential energy of about 1900 J. For simplification, the RMW (r_m) of the intensifying storm in the dynamical efficiency calculation [cf. Eq. (9)] is assumed tos be constant and equal to r_mpi. See text for further references.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0393.1

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For both n = 0.5 (red curves) and 1.0 (green curves), the IP at low TC intensity is considerably reduced because of the low dynamical efficiency due to weak inner-core inertial stability. In both cases, the IR exhibits a maximum at an intermediate intensity. For n = 0.5, the maximum IR reaches about 120 m s⁻¹ day⁻¹ at a maximum near surface wind speed of about 25 m s⁻¹, while for n = 1.0, the maximum IR reaches 60 m s⁻¹ day⁻¹ at V_max = 30 m s⁻¹. Thus, the modified conceptual model given by Eq. (12) provides a reasonable upper limit to the maximum IR with n = 1.0 and it captures the overall intensity dependence of the TC IR as found in observations (cf. Fig. 1). This strongly suggests that the introduction of the dynamical efficiency to the simplified energetically based dynamical system model of TC intensification is physically meaningful and thus acceptable to reproduce the intensity dependence of TC IR in agreement with that from observational studies (Kaplan et al. 2010; Xu and Wang 2015, 2018a).

To find the TC intensity where the IR reaches its maximum, i.e., the maximum potential IR (MPIR), we differentiate Eq. (12) with respect to V_max. Assuming C_D and H are constant, the IR of the TC reaches a maximum when

${\displaystyle \frac{\partial }{\partial V_{\max }}}\left({\displaystyle \frac{d{V}_{\max }}{dt}}\right)={\displaystyle \frac{C_D}{H}}\left(V_{\text{mpi}}^2\hspace{0.17em}{\displaystyle \frac{\partial E}{\partial V_{\max }}}-2V_{\max }\right)=0,$

namely,

$V_{\text{mpir}}={\displaystyle \frac{V_{\text{mpi}}^2}{2}}{\displaystyle \frac{\partial E}{\partial V_{\max }}}.$(13)

Note, in deriving Eq. (13) from Eq. (12), the possible dependences of the thermodynamic efficiency, the surface exchange coefficients, and the enthalpy gradient across the air–sea interface on the surface wind speed near r_m (which are all implicitly included in the MPI) have been ignored. We argue that this is a reasonable approximation in the context of a conceptual dynamical system model since the effect is often quite small (cf. Fig. 1 in Emanuel 2012). Equation (12) indicates that the intensity-dependent nature of the TC IR depends strongly on the power index n related to the dynamical efficiency E of the TC system. Figure 3 shows the numerical solution of the dependence of the maximum wind speed V_max at which the IR reaches its maximum (V_mpir) on the power index n. As n increases, the MPIR decreases and occurs at relatively higher maximum wind speeds. Comparing Figs. 1 and 2, we can find that an n value around 1.0 is reasonable for the chosen H and other parameters to a large extent. In the case shown, the MPIR can reach up to 60 m s⁻¹ day⁻¹ at a maximum wind speed around 30 m s⁻¹. Hence, the MPIR provides a reasonable upper limit to the maximum IR (Fig. 1), in a similar way to the MPI, which gives an upper limit to the maximum potential intensity, given favorable atmospheric and oceanic thermodynamic conditions.

Maximum intensification rate (IR_max, m s⁻¹ day⁻¹, solid curve, left Y axis) and corresponding storm intensity (V_max, m s⁻¹, dashed curve, right Y axis) as a function of power index n, which sets the dependency degree of the IP on dynamical efficiency E defined by the normalized inertial stability frequency. Other parameters are the same as those chosen in the calculations for Fig. 2.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0393.1

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Maximum intensification rate (IR_max, m s⁻¹ day⁻¹, solid curve, left Y axis) and corresponding storm intensity (V_max, m s⁻¹, dashed curve, right Y axis) as a function of power index n, which sets the dependency degree of the IP on dynamical efficiency E defined by the normalized inertial stability frequency. Other parameters are the same as those chosen in the calculations for Fig. 2.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0393.1

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Maximum intensification rate (IR_max, m s⁻¹ day⁻¹, solid curve, left Y axis) and corresponding storm intensity (V_max, m s⁻¹, dashed curve, right Y axis) as a function of power index n, which sets the dependency degree of the IP on dynamical efficiency E defined by the normalized inertial stability frequency. Other parameters are the same as those chosen in the calculations for Fig. 2.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0393.1

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Kaplan et al. (2010) hypothesized that the turning point from a positive to a negative correlation between the TC intensity and the corresponding IR (as shown in Fig. 1) may be associated with a balance between the thermodynamic/dynamic efficiency in response to eyewall heating and the potential of the TC to intensify toward its MPI. Our modified energetically based dynamical system model above quantifies this hypothesis and can reproduce the observed intensity dependence of the TC IR reasonably well. Therefore, with the introduction of the dynamical efficiency as a factor in the IP of the TC system, the energetically based TC intensification model has potential to be used for real TC intensity forecasts as it captures the major features of the intensity dependence of the TC IR as evident from observational data. Note that, from a forecast perspective, both the steady-state intensity (MPI) and RMW in the model should be estimated in advance. In addition, the dynamical efficiency discussed in this study can be considered the intrinsic dynamical efficiency of the TC vortex. In reality, many other factors may limit the dynamical efficiency, such as the environmental vertical wind shear and translation of the TC itself. However, as our first effort, we have only focused on the intrinsic dynamical efficient in this study.

3. Results from idealized full-physics model simulations

The modified energetically based dynamical system model for the TC IR outlined in section 2 is heavily conceptualized. To examine the extent to which the simplified model is valid, a series of ensemble numerical experiments are performed using the state-of-the-art axisymmetric Cloud Model 1 (CM1; Bryan and Fritsch 2002). Considering the TC IR varies with SST (Fig. 1), to ensure the robustness of the main results, experiments with different SST are performed. The experimental design follows the SST-dependent sounding experiments in Li et al. (2020) to ensure the consistency between SST and initial atmospheric sounding as in nature. There are four ensemble experiments with different SST and corresponding atmospheric sounding sorted for the WNP (labeled as SST28_WNP–SST31_WNP), and four ensemble experiments those sorted for the North Atlantic (NA; labeled as SST27_NA–SST30_NA). Each ensemble experiment has 21 runs. In the standard run the maximum tangential wind speed of the initial TC vortex is 15 m s⁻¹ at the surface with the RMW of 80 km, and in the other 20 runs the maximum tangential wind speed and the RMW are perturbed by ±0.1 m s⁻¹ (for 10 runs) and ±0.4 km (for 10 runs), respectively. All experiments are run on an f plane of 20°N in a quiescent environment with the model domain of 3100 km × 25 km in the radial and vertical directions, respectively. The radial grid spacing is 1 km within 100-km radius and is stretched to 12 km at the lateral boundary. The vertical grid spacing is stretched below 5.5 km with a total of 59 vertical levels used for the model atmosphere. The microphysics scheme of Thompson et al. (2008) is used for cloud microphysical processes and cumulus convection is not considered. Newtonian cooling, capped at 2 K day⁻¹, is used to mimic radiative cooling (Rotunno and Emanuel 1987), and dissipative heating is not considered. The Smagorinsky scheme (Bryan and Fritsch 2002) is used to mimic subgrid-scale turbulent mixing with the horizontal and asymptotic vertical mixing lengths being fixed at 700 and 70 m as in Li et al. (2020), respectively. The wind-dependent surface drag coefficient (C_D) of Donelan et al. (2004), which increases with wind speed at low wind speeds (<25 m s⁻¹) but keeps a constant (2.4 × 10⁻³) at high wind speeds, is used for surface stress calculation, while a constant exchange coefficient (C_k) of 1.2 × 10⁻³ is used for surface enthalpy flux calculation. All simulations are integrated for 240 h with hourly model outputs, and we only used the intensity evolution from the ensemble mean for each experiment of simulations in our following analyses to avoid effects of convective processes and cloud-scale motions.

Figure 4 shows the time evolution of the maximum 10-m wind speed (V_max) and the corresponding RMW of each of the ensemble simulations and their corresponding ensemble mean for all experiments. All storms started to intensify after an initial adjustment period, and the adjustment period decreases with increasing SST (Figs. 4a,b). For each experiment, the maximum IR occurs when the maximum storm intensity is between ~30 and 40 m s⁻¹ similar to that found in observations (Fig. 1). For both basins, the maximum IR increases with increasing SST as in observations (Fig. 1). The RMW undergoes a continuous contraction after an initial adjustment period. This continuous contraction of the RMW coincides with the early and rapid intensification of the simulated storms. Afterward, the RMW decreases very slowly and the contraction stops well before the end of intensification in each experiment. This behavior is consistent with recent findings from both observations and idealized numerical simulations by Stern et al. (2015) and Li et al. (2019), who also found that the contraction of the eyewall and the RMW does not necessarily coincide with the RI period. Note that although some transient variability occurs in the IR among the ensemble simulations, the overall trends are quite similar. Therefore, since our major interest is in the overall intensification of the simulated TCs, we only focus on the ensemble mean results instead of displaying each of the 21 simulations for each experiment in the following discussion.

Time evolution of (a),(b) maximum 10-m wind speeds and (c),(d) corresponding radii of maximum wind in ensemble simulations (thin curves) and their corresponding ensemble averages (thick curves) for experiments using the SST-sorted atmospheric soundings (a),(c) over the western North Pacific and (b),(d) over the North Atlantic.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0393.1

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Time evolution of (a),(b) maximum 10-m wind speeds and (c),(d) corresponding radii of maximum wind in ensemble simulations (thin curves) and their corresponding ensemble averages (thick curves) for experiments using the SST-sorted atmospheric soundings (a),(c) over the western North Pacific and (b),(d) over the North Atlantic.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0393.1

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Time evolution of (a),(b) maximum 10-m wind speeds and (c),(d) corresponding radii of maximum wind in ensemble simulations (thin curves) and their corresponding ensemble averages (thick curves) for experiments using the SST-sorted atmospheric soundings (a),(c) over the western North Pacific and (b),(d) over the North Atlantic.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0393.1

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Based on Emanuel (2012) and Emanuel and Zhang (2017), the height parameter H was roughly twice the depth of the boundary layer. Considering the fact that the characteristic height scale of the TC inner-core boundary layer, no matter in dynamic or thermodynamic definition, is usually between ~500 and 2000 m (Zhang et al. 2011), the height parameter H can be chosen between ~1000 and 4000 m. Figure 5 shows the scatter diagram of the diagnosed dynamical efficiency from Eq. (17) under different H based on the ensemble-mean results of all experiment, with three different forms of dynamical efficiency shown, i.e., in terms of the normalized inertial stability (E) in Eq. (7), the normalized simplified inertial stability (E_sim) in Eq. (9), and the normalized absolute vertical vorticity (E²/E_sim). For each H, the distributions among the three different forms of dynamical efficiency are similar, especially between E and E_sim (first and second columns in Fig. 5), implying that the normalized simplified inertial stability, Eq. (9), is a good approximation of the normalized inertial stability, Eq. (7), to parameterize the dynamical efficiency. From Fig. 5, for H = 2000 m, we can find that for all the three forms of dynamical efficiency, all the diagnostic results are smoothly distributed around the line of n = 1. However, with the varying H, more nonconstant power indices (n) are needed to capture the overall diagnostic results. Therefore, the settings H = 2000 m and n = 1 can be used to ensure the convergence of the diagnostic results. Because all required data are available from the CM1 output for all simulations, the dynamical efficiency is calculated using its original form, Eq. (7), with the results shown in Fig. 6 for a reference. Note that the results discussed below are similar to those calculated with the simplified form of the dynamical efficiency, Eq. (9). In addition, from the green dotted line in Fig. 2, n = 1 gives a linear relationship between IP and storm intensity in terms of maximum 10-m wind speed, consistent with the nearly linear relationship between the dynamical efficiency and the TC intensity from the numerical experiments (Fig. 6). This nearly linear relation is in fairly agreement with results from previous studies based on balanced dynamics (Schubert and Hack 1982, 1983; Pendergrass and Willoughby 2009). Note that the variation of the RMW with time has been included in the calculation of dynamical efficiency here (Fig. 6), which is different from that assumed in Fig. 2. This partly explains why the relationship between the dynamical efficiency and the TC intensity in Fig. 6 is not exactly linear as shown in Fig. 2.

Scatter diagram of the diagnosed dynamical efficiency based on Eq. (17) against (a),(d),(g),(j) normalized inertial stability in Eq. (7), (b),(e),(h),(k) normalized simplified inertial stability in Eq. (9), and (c),(f),(i),(l) normalized absolute vorticity for (a)–(c) H = 1000 m, (d)–(f) H = 2000 m, (g)–(i) H = 3000 m, and (j)–(l) H = 4000 m.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0393.1

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Scatter diagram of the diagnosed dynamical efficiency based on Eq. (17) against (a),(d),(g),(j) normalized inertial stability in Eq. (7), (b),(e),(h),(k) normalized simplified inertial stability in Eq. (9), and (c),(f),(i),(l) normalized absolute vorticity for (a)–(c) H = 1000 m, (d)–(f) H = 2000 m, (g)–(i) H = 3000 m, and (j)–(l) H = 4000 m.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0393.1

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Scatter diagram of the diagnosed dynamical efficiency based on Eq. (17) against (a),(d),(g),(j) normalized inertial stability in Eq. (7), (b),(e),(h),(k) normalized simplified inertial stability in Eq. (9), and (c),(f),(i),(l) normalized absolute vorticity for (a)–(c) H = 1000 m, (d)–(f) H = 2000 m, (g)–(i) H = 3000 m, and (j)–(l) H = 4000 m.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0393.1

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The dynamical efficiency E estimated using Eq. (7) as a function of (a),(c) time and (b),(d) the simulated ensemble-mean maximum 10-m wind speed based on the ensemble mean results for experiments using the SST-sorted atmospheric soundings (a),(b) over the western North Pacific and (c),(d) over the North Atlantic.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0393.1

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The dynamical efficiency E estimated using Eq. (7) as a function of (a),(c) time and (b),(d) the simulated ensemble-mean maximum 10-m wind speed based on the ensemble mean results for experiments using the SST-sorted atmospheric soundings (a),(b) over the western North Pacific and (c),(d) over the North Atlantic.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0393.1

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The dynamical efficiency E estimated using Eq. (7) as a function of (a),(c) time and (b),(d) the simulated ensemble-mean maximum 10-m wind speed based on the ensemble mean results for experiments using the SST-sorted atmospheric soundings (a),(b) over the western North Pacific and (c),(d) over the North Atlantic.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0393.1

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Figures 7 and 8 compare the simulated IR (IR_sim) given by the time tendency of maximum 10-m wind speed from the ensemble mean of the full-physics axisymmetric model simulated V_max, and that estimated from the energetically based dynamical system IR model, Eq. (17), from all experiments, with the dynamical efficiency given by Eq. (7) and all other variables from the ensemble mean simulations, except n and H, which are chosen as discussed above. For all experiments, the estimated IR using the modified energetically based dynamical system model can capture well the evolution of the simulated IR. In particular, the energetically based IR model delivers good approximations to both the maximum IR and the intensity dependence of the IR in the simulations as in observations mentioned in section 3. The simplified dynamical system IR model gives a maximum IR between 30 and 50 m s⁻¹ day⁻¹ around the intermediate intensity for each experiment, which agrees well with the model simulations and also fairly consistent with observations (Fig. 1). For both basins, both the estimated maximum IR and the intensity at the maximum IR (V_mpir) tend to increase with increasing SST, which is also in agreement with the model simulations (Figs. 7 and 8) and observations (Fig. 1).

(a),(c),(e),(g) Time evolution of the simulated ensemble-mean maximum 10-m wind speed tendency (IR_sim, m s⁻¹ day⁻¹, orange curve) and the estimated intensification rate according to the energetic intensification rate by Eq. (17) (IR_est, m s⁻¹ day⁻¹, cyan curve) from experiments using the SST-sorted atmospheric soundings over the western North Pacific, and (b),(d),(f),(h) IR_sim (m s⁻¹ day⁻¹, orange curve) and IR_est (m s⁻¹ day⁻¹, cyan curve) as a function of the simulated ensemble-mean maximum 10-m wind speed (x axis: V_max in m s⁻¹). Besides H and n, all variables used to calculate the estimated IR are taken from the ensemble-mean simulations. The dynamical efficiency E is estimated using Eq. (7). See text for further details.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0393.1

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(a),(c),(e),(g) Time evolution of the simulated ensemble-mean maximum 10-m wind speed tendency (IR_sim, m s⁻¹ day⁻¹, orange curve) and the estimated intensification rate according to the energetic intensification rate by Eq. (17) (IR_est, m s⁻¹ day⁻¹, cyan curve) from experiments using the SST-sorted atmospheric soundings over the western North Pacific, and (b),(d),(f),(h) IR_sim (m s⁻¹ day⁻¹, orange curve) and IR_est (m s⁻¹ day⁻¹, cyan curve) as a function of the simulated ensemble-mean maximum 10-m wind speed (x axis: V_max in m s⁻¹). Besides H and n, all variables used to calculate the estimated IR are taken from the ensemble-mean simulations. The dynamical efficiency E is estimated using Eq. (7). See text for further details.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0393.1

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(a),(c),(e),(g) Time evolution of the simulated ensemble-mean maximum 10-m wind speed tendency (IR_sim, m s⁻¹ day⁻¹, orange curve) and the estimated intensification rate according to the energetic intensification rate by Eq. (17) (IR_est, m s⁻¹ day⁻¹, cyan curve) from experiments using the SST-sorted atmospheric soundings over the western North Pacific, and (b),(d),(f),(h) IR_sim (m s⁻¹ day⁻¹, orange curve) and IR_est (m s⁻¹ day⁻¹, cyan curve) as a function of the simulated ensemble-mean maximum 10-m wind speed (x axis: V_max in m s⁻¹). Besides H and n, all variables used to calculate the estimated IR are taken from the ensemble-mean simulations. The dynamical efficiency E is estimated using Eq. (7). See text for further details.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0393.1

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As in Fig. 7, but for experiments using the SST-sorted atmospheric soundings over the North Atlantic.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0393.1

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As in Fig. 7, but for experiments using the SST-sorted atmospheric soundings over the North Atlantic.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0393.1

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As in Fig. 7, but for experiments using the SST-sorted atmospheric soundings over the North Atlantic.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0393.1

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To further ensure the robustness of the dynamical system model and the selections of n and H, several additional sensitivity ensemble experiments were performed. Those sensitivity experiments are the same as SST28_WNP, but with the latitude of f plane decreased to 10°N (labeled as SST28_WNP_10N) and increased to 30°N (labeled as SST28_WNP_30N), with the RMW of the initial TC vortex decreased to 60 km (labeled as SST28_WNP_60km) and increased to 100 km (labeled as SST28_WNP_100km), with the enthalpy exchange coefficient halved (labeled as SST28_WNP_CK05) and doubled (labeled as SST28_WNP_CK20), with the horizontal mixing length halved (labeled as SST28_WNP_lh05) and doubled (labeled as SST28_WNP_lh20), and with the asymptotic vertical mixing length halved (labeled as SST28_WNP_lv05) and doubled (labeled as SST28_WNP_lv20). The simulated IR and estimated IR from Eq. (17) with H = 2000 m and n = 1 from those sensitivity experiments are shown in Fig. 9. As expected, for all experiments, the estimated IR can capture well the evolution of the simulated IR, and gives a maximum IR at the intermediate intensity as in the model simulations. Considering the simplicity of the energetically based dynamical system model, the discrepancies between the simulated IR and the estimated IR from Eq. (17) seem to be acceptable. The simplified energetically based IR model ignores any detailed dynamics that may cause short-term variability in intensity change. Even though the IR behavior related to small-scale motions is not considered, the IR equation proposed in this study is able to deliver a reasonable estimate for the upper IR limit, and the overall IR dependency on TC intensity as found in observations.

As in Fig. 7b, but from different sensitivity experiments as indicated by the legends.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0393.1

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As in Fig. 7b, but from different sensitivity experiments as indicated by the legends.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0393.1

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As in Fig. 7b, but from different sensitivity experiments as indicated by the legends.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0393.1

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4. Conclusions and discussion

A pronounced feature from recent statistical analyses based on best track TC data is that the maximum IR is observed most frequently when the maximum sustained surface wind speed of a TC is at an intermediate intensity of about 30–40 m s⁻¹ (Kaplan et al. 2010; Xu and Wang 2015, 2018a). Such an intensity dependence of the TC IR was previously explained qualitatively in terms of a core heating efficiency that enhances with the storm’s intensity due to higher inner-core inertial stability on one hand, and less room for the storm to further intensify as it approaches its MPI on the other hand (Kaplan et al. 2010). Moreover, in several studies, time-dependent equations for the maximum near-surface wind speed (DeMaria 2009; Emanuel 1997, 2012; Ozawa and Shimokawa 2015; Emanuel and Zhang 2017) were derived. Although the logistical growth equation-based model of DeMaria (2009) and the modified theoretical model of Emanuel and Zhang (2017) from Emanuel (2012) could capture the overall evolution of TC intensity, the intensity dependence of the TC IR was not discussed in these studies, which mainly focused on the environmental factors rather than the TC intrinsic dynamical parameters.

In this study, an empirical dynamical efficiency has been introduced to the simple energetically based model recently proposed by Ozawa and Shimokawa (2015). This simple model is based conceptually on the viewpoint that a TC can be described energetically as a Carnot heat engine in a non-steady-state framework (Wang 2012, 2015). Here we have augmented this concept by introducing an empirical dynamical efficiency factor, which is presumed to be a function of the inner-core inertial stability, and to estimate the inner-core heating efficiency along with earlier theoretical studies based on balanced dynamics of TC intensification (Schubert and Hack 1982, 1983; Pendergrass and Willoughby 2009). With the introduced dynamical efficiency, the modified energetically based dynamical system model can explain and quantify the observed intensity dependence of the TC IR fairly well.

According to the modified simple energetically based IR model, the TC IR is controlled by the intensification potential and the weakening rate due to surface friction beneath the eyewall. The intensification potential is determined primarily by the potential energy available for a TC to develop, which is a function of the thermodynamic conditions of the atmosphere and the underlying ocean, and the dynamical efficiency of the TC system. The latter factor, which we introduced in this study, depends strongly on the degree of convective organization within the eyewall and the inner-core inertial stability of the storm. At relatively weak intensity, the intensification potential of the intensifying storm is larger than the frictional weakening rate, leading to an increase in the TC IR with TC intensity in the initial stage. As the storm reaches an intermediate intensity of about 30–40 m s⁻¹, the difference between the intensification potential and frictional weakening rate reaches its maximum, concurrent with the maximum IR, or “maximum potential IR (MPIR),” of the TC. Later on, as the TC intensifies further, the IR decreases because surface friction increases with TC intensity at a faster rate than the intensification potential. Finally, as the storm approaches its MPI, the IR becomes zero.

The modified energetically based IR model is further validated with results from idealized ensemble full-physics model simulations with a nonhydrostatic, convection-resolving TC model. The full-physics model simulations reproduce the observed intensity dependence of the TC IR reasonably well. Results from the estimated IR using the energetically based IR equation show that the net energy gain beneath the eyewall fits well the IR of the simulated storm. It is also shown that the modified simplified energetically based IR model reproduces the overall IR of the simulated storm. Although some discrepancies exist between the IR in the full-physics model simulations and that estimated from the simplified energetically based IR model due to its lack of any detailed dynamics, the simplified energetically based IR model may provide an estimate of the IR of a TC in its transient stage. It captures the major features of the intensity dependence of the TC IR and its upper limit as evident from observations. Moreover, since the maximum potential IR is proportional to the MPI of the TC and the TC MPI is a function of SST and other environmental parameters, we may expect a trend in MPIR in response to environmental changes similar to that found for MPI (e.g., DeMaria and Kaplan 1994; Emanuel 1988). Hence, with view to many studies that have indicated possible increase in the MPI in the projected future climate, the modified simple energetically based dynamical system model of TC IR implies a correspondingly potential increase in TC IR in response to global warming as discussed by Emanuel (2017).

Finally, we should mention that in this study we have not considered any detrimental effects from unfavorable environmental conditions on TC IR. Therefore, the dynamical efficiency introduced in this study is purely an ad hoc parameterization related to the inner-core dynamics of the TC itself. In this sense, we can consider the inner-core inertial stability determined dynamical efficiency as the intrinsic dynamical efficiency. This intrinsic dynamical efficient is the maximum efficiency of real TCs and thus the simplified energetically based IR model can provide the maximum possible IR at all stages of an intensifying TC. In our future studies, we will further introduce factors that may reduce the dynamical efficiency of the TC system, such as environmental vertical wind shear, the TC-induced ocean cooling, the convective activity outside the inner core, and so on. In addition, additional effort is under way to estimate parameters/constants in the simplified energetically based IR model using the best track TC data and statistical analysis, including machine learning algorithm. The results will be reported separately in due course.