## 1. Introduction

Tang and Emanuel (2010, hereafter TE10) developed a theory for both the maximum potential intensity and the evolution of storm intensity, as measured by peak storm wind speed, that accounts for the effects of the import of midlevel environmental low-entropy air into the storm (termed ventilation) by environmental vertical wind shear. Tang and Emanuel (2012, hereafter TE12) then applied this theory to the historical record of observations and demonstrated that the theory successfully predicts the historical statistics of the dynamics of storm intensity within a phase space defined by two parameters: 1) the current intensity normalized by the maximum potential intensity and 2) a “ventilation index,” which is a nondimensional quantity capturing the combined effects of midlevel dry air and wind shear for a given local thermodynamic environment. This ventilation index was further demonstrated by TE12 to be a skillful indicator of genesis potential, a finding that has since been corroborated by Tang and Camargo (2014) and Camargo et al. (2014) for the current climate and under future climate change, as well as by Korty et al. (2012) for simulations of the Last Glacial Maximum and by Yan et al. (2015) for the last two millennia.

Implicit in TE10’s theory for intensity dynamics is the role of the enhancement of surface enthalpy (or entropy) fluxes at higher wind speeds, commonly termed the wind-induced surface heat exchange (WISHE) feedback (Emanuel 1986). However, recent numerical simulation studies (Montgomery et al. 2009, 2015) have questioned the necessity of the WISHE feedback in intensifying a tropical cyclone and/or maintaining one at a steady state, arguing that the simulation of intensifying storms in the presence of a capped surface flux wind speed undermines the validity of extant tropical cyclone theory. In response to this issue, Zhang and Emanuel (2016) employed a classical model for linear instability as well as a set of numerical tropical cyclone simulation experiments with capped surface enthalpy fluxes to argue that this feedback is important to the time-dependent evolution of tropical cyclone intensity in the real world even if it is not strictly necessary to the existence of the underlying instability itself.

Here we complement the work of Zhang and Emanuel (2016) by generalizing the observationally validated theory of TE10 and TE12 to a scenario with capped surface entropy fluxes. To do so, we first provide a simplified rederivation of the theoretical result of TE12 and provide novel analytical solutions for various aspects of the phase-space solution (section 2). We then generalize this result to account for capped surface entropy fluxes (section 3) and analyze the effect of capping the surface fluxes on the predicted dynamics of storm intensity both within the same TE12 phase space normalized by the traditional potential intensity (section 3a), and in a phase space normalized by the capped-flux potential intensity (section 3b). We conclude with a summary and discussion of key conclusions (section 4).

## 2. Ventilation theory

### a. Reproducing TE12 theory

We begin with a simple and direct derivation of the final theoretical result of TE10 from a three-term power budget equation. This derivation was implicit in the supplement of TE12, though to the author’s knowledge an explicit derivation does not currently exist in the literature.

*η*is the thermodynamic efficiency of the heat engine system. The thermodynamic efficiency in Eq. (1) is classically defined as the Carnot efficiency

*ρ*is the density of boundary layer air,

*υ*is the total near-surface wind speed,

*s*is the entropy of the overlying near-surface air. The power sink due to frictional dissipation is given by

*υ*may be interpreted simply as the maximum wind speed (i.e., intensity) of the storm. Allowing for the entropy transferred to the surface via frictional dissipation to be recycled back into the boundary layer introduces a multiplicative factor

*c*is a constant [cf. TE12, their Eqs. (S6)–(S8)].

^{1}

*υ*at equilibrium

### b. Analytical solutions for intensity dynamics

TE12 provided a graphical phase-space solution for

## 3. Generalization to capped surface entropy flux wind speed

### a. Standard potential intensity nondimensionalization

We now generalize the above theory to the case where the wind speed dependence of surface entropy fluxes is capped at an upper-bound wind speed

The above analytical solution, equilibria, and extrema are displayed in Fig. 2 within the same phase space shown in Fig. 1 but now over a range of values of

As in Fig. 1 for

Citation: Journal of the Atmospheric Sciences 74, 9; 10.1175/JAS-D-17-0061.1

As in Fig. 1 for

Citation: Journal of the Atmospheric Sciences 74, 9; 10.1175/JAS-D-17-0061.1

As in Fig. 1 for

Citation: Journal of the Atmospheric Sciences 74, 9; 10.1175/JAS-D-17-0061.1

### b. Capped-flux potential intensity nondimensionalization

The above solution is displayed in Fig. 3 for the same range of values of external parameter

As in Fig. 2, but with the system nondimensionalized by

Citation: Journal of the Atmospheric Sciences 74, 9; 10.1175/JAS-D-17-0061.1

As in Fig. 2, but with the system nondimensionalized by

Citation: Journal of the Atmospheric Sciences 74, 9; 10.1175/JAS-D-17-0061.1

As in Fig. 2, but with the system nondimensionalized by

Citation: Journal of the Atmospheric Sciences 74, 9; 10.1175/JAS-D-17-0061.1

For further analysis, we note that three regimes exist in this system. First, for *υ* (i.e., *υ* for

More broadly, Fig. 3 demonstrates that the qualitative character of the solution, including both intensity changes and intensity equilibria, phrased relative to the capped-flux potential intensity, is quite similar to that of the original solution of Fig. 1. This indicates that the dynamics of storm intensity are not qualitatively different when the surface entropy flux wind speed is capped compared to when it is not (i.e., in the real world), though changes in storm intensity are now defined relative to a lower, capped-flux potential intensity. This finding corroborates that of Zhang and Emanuel (2016), which demonstrated that the wind speed dependence of the surface heat flux is an important component of the intensity dynamics of real-world storms even if it is not a necessary condition for intensification of storms in general.

Note that, for the regime detailed above, the factor

## 4. Discussion and conclusions

We have presented a simple derivation of the core outcome of the tropical cyclone ventilation theory developed by TE10, which was demonstrated by TE12 to reproduce the statistics of the dynamics of tropical cyclone intensity in the historical record. We then extended this derivation to analytical solutions for the complete phase space describing the dynamics of storm intensity. Finally, we generalized this derivation to the case of capped surface entropy fluxes and phrased the solutions relative to both the traditional potential intensity and a capped-flux potential intensity. We find that the essential dynamics of storm intensity are qualitatively identical with or without a cap on the surface entropy flux wind speed; these dynamics are simply defined relative to a new potential intensity that is reduced by the flux capping. Specifically, then, these results indicate that, for sufficiently small values of ventilation, an intensifying storm in the presence of a capped surface entropy flux wind speed would not only not be surprising but would be predicted by theory. Conceptually, this behavior occurs simply because there can exist a residual power surplus available to intensify the storm even when the surface fluxes become capped and the WISHE feedback ceases. This finding aligns with the numerical simulation outcomes of Zhang and Emanuel (2016) in two key respects. First, the theory demonstrates the WISHE feedback is an important component of the intensity dynamics of real-world storms, yet it is not a necessary condition for intensification of storms in general. Second, because a surface flux wind speed cap acts to effectively enhance the detrimental effect of ventilation, the theory predicts that at sufficiently high ventilation a storm may intensify in the presence of WISHE but decay in its absence. Thus, taken together, these results indicate that there is no fundamental disconnect between extant tropical cyclone theory and the finding in numerical simulations that a storm may intensify in the presence of capped surface entropy fluxes. Future work might seek to test the theoretical predictions presented here in carefully designed numerical simulation experiments, which could further provide precise quantitative evidence for these conclusions.

## Acknowledgments

The author thanks Tim Cronin for the valuable discussion on the nature of entropy fluxes. The author thanks Brian Tang and two anonymous reviewers for their feedback in improving this manuscript. All data and code in this analysis are publicly available upon request from the corresponding author.

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