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  • Bister, M., N. Renno, O. Pauluis, and K. Emanuel, 2011: Comment on Makarieva et al. “A critique of some modern applications of the Carnot heat engine concept: the dissipative heat engine cannot exist?” Proc. Roy. Soc. London, A467, 16, doi:10.1098/rspa.2010.0087.

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  • Camargo, S. J., M. K. Tippett, A. H. Sobel, G. A. Vecchi, and M. Zhao, 2014: Testing the performance of tropical cyclone genesis indices in future climates using the HIRAM model. J. Climate, 27, 91719196, doi:10.1175/JCLI-D-13-00505.1.

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  • Chen, S. S., W. Zhao, M. A. Donelan, and H. L. Tolman, 2013: Directional wind–wave coupling in fully coupled atmosphere–wave–ocean models: Results from CBLAST-Hurricane. J. Atmos. Sci., 70, 31983215, doi:10.1175/JAS-D-12-0157.1.

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  • Korty, R. L., S. J. Camargo, and J. Galewsky, 2012: Tropical cyclone genesis factors in simulations of the last glacial maximum. J. Climate, 25, 43484365, doi:10.1175/JCLI-D-11-00517.1.

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  • View in gallery

    Analytical solution [Eq. (18)] for joint dependence of normalized intensity change on normalized intensity () and normalized ventilation . Purple line denotes equilibrium solution [Eq. (19)], including stable branch (solid), corresponding to the ventilation-dependent maximum potential intensity, and unstable branch (dashed), corresponding to the ventilation-dependent genesis threshold. Markers: × denotes the largest possible value of that can support a storm at equilibrium [Eq. (20)]; * denotes location of maximum normalized intensification rate [Eq. (21)] with value of annotated. For negative values of whose magnitude exceeds (i.e., storm lysis), is simply set equal to .

  • View in gallery

    As in Fig. 1 for , but for the case of capped surface entropy fluxes [Eqs. (24) and (25)]. Subplots show the solution for . Gray line indicates normalized capping surface entropy flux wind speed . Circle denotes . Setting to be large fully recovers the result of Fig. 1.

  • View in gallery

    As in Fig. 2, but with the system nondimensionalized by in lieu of [i.e., ; ; Eqs. (30) and (31)]. Subplots show the solution for the same range of values of as Fig. 2. Gray line indicates . Setting to be large fully recovers the result of Fig. 1.

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A Simple Derivation of Tropical Cyclone Ventilation Theory and Its Application to Capped Surface Entropy Fluxes

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  • 1 Purdue University, West Lafayette, Indiana
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Abstract

In a recent study, a theory was presented for the dependence of tropical cyclone intensity on the ventilation of dry air by environmental vertical wind shear. This theory was found to successfully capture the statistics of intensity dynamics in the historical record. This theory is rederived here from a simple three-term power budget and extended to analytical solutions for the complete phase space, including the change in storm intensity itself. The derivation is then generalized to the case of a capped surface entropy flux wind speed, including analytical solutions defined relative to both the traditional potential intensity and the capped-flux potential intensity. The results demonstrate that a cap on the surface entropy flux wind speed reduces the potential intensity of the system and effectively amplifies the detrimental effect of ventilation on the tropical cyclone heat engine. However, such a cap does not alter the qualitative structure of the phase-space solution for intensity change phrased relative to the capped-flux potential intensity. Thus, the wind speed dependence of surface entropy fluxes is important for intensity change in real-world storms, though it is not a necessary condition for intensification in general. Indeed, a residual power surplus may remain available to intensify a storm even in the presence of a cap, though intensification may be fully suppressed for sufficiently strong ventilation. This work complements a recent numerical simulation study and provides further evidence that there is no 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.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Daniel R. Chavas, drchavas@gmail.com

Abstract

In a recent study, a theory was presented for the dependence of tropical cyclone intensity on the ventilation of dry air by environmental vertical wind shear. This theory was found to successfully capture the statistics of intensity dynamics in the historical record. This theory is rederived here from a simple three-term power budget and extended to analytical solutions for the complete phase space, including the change in storm intensity itself. The derivation is then generalized to the case of a capped surface entropy flux wind speed, including analytical solutions defined relative to both the traditional potential intensity and the capped-flux potential intensity. The results demonstrate that a cap on the surface entropy flux wind speed reduces the potential intensity of the system and effectively amplifies the detrimental effect of ventilation on the tropical cyclone heat engine. However, such a cap does not alter the qualitative structure of the phase-space solution for intensity change phrased relative to the capped-flux potential intensity. Thus, the wind speed dependence of surface entropy fluxes is important for intensity change in real-world storms, though it is not a necessary condition for intensification in general. Indeed, a residual power surplus may remain available to intensify a storm even in the presence of a cap, though intensification may be fully suppressed for sufficiently strong ventilation. This work complements a recent numerical simulation study and provides further evidence that there is no 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.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Daniel R. Chavas, drchavas@gmail.com

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.

A tropical cyclone at steady state may be viewed as an atmospheric heat engine that converts thermal energy into mechanical (kinetic) energy (Emanuel 1986; Bister et al. 2011; Pauluis 2011): that is,
e1
where is the thermal energy input into the system, is the mechanical energy output, and η is the thermodynamic efficiency of the heat engine system. The thermodynamic efficiency in Eq. (1) is classically defined as the Carnot efficiency , where is the temperature of heat input at the warm surface, and is the temperature of heat loss due to radiative cooling in the cold outflow aloft for a trajectory originating at the radius of maximum wind (Emanuel 1986); this formulation assumes that the eyewall is saturated (Pauluis 2011). This energy conversion may be more conveniently formulated in terms of entropy input into the system to power the storm circulation against frictional dissipation: that is,
e2
where is the net entropy source and is the power sink due to surface frictional dissipation. The dominant entropy source for the system is due to surface entropy fluxes, given by
e3
where is the entropy flux into the boundary layer from the surface, is a bulk surface entropy exchange coefficient, ρ is the density of boundary layer air, υ is the total near-surface wind speed, is the saturation entropy of the sea surface, and s is the entropy of the overlying near-surface air. The power sink due to frictional dissipation is given by
e4
where is a bulk surface momentum exchange (drag) coefficient. Both surface fluxes are assumed to be dominated by their values at the radius of maximum wind, and thus 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 to the LHS of Eq. (2), which effectively acts to increase the thermodynamic efficiency to (Bister and Emanuel 1998); caveats of this accounting are discussed in Kieu (2015). A second sink of power for the system arises because of the “ventilation” of low-entropy (primarily dry) midlevel environmental air into the boundary layer at the radius of maximum wind by environmental vertical wind shear. This sink is best considered as a negative source term, though, as it acts as a direct reduction to the source term ; for this reason it was termed “anti-fuel” in TE12. It may also reduce the thermodynamic efficiency by increasing . This negative source may be approximated as
e5
where is the magnitude of the bulk wind shear in the background steering flow (typically the 850–200-hPa pressure layer), is the midlevel (typically the 600-hPa level) saturation entropy in the convecting inner core, is the midlevel entropy in the environment outside of the inner core, and c is a constant [cf. TE12, their Eqs. (S6)–(S8)].1
We may generalize Eq. (2) to the unsteady case, which yields a simple three-term power budget equation given by
e6
where represents the residual power surplus/deficit for the system in its current state. When positive, this term equals the surplus power available to do additional mechanical work against frictional dissipation; when negative, this term equals the power deficit between that which is available to do mechanical work and that which is lost as a result of frictional dissipation. Substituting Eqs. (3)(5) into Eq. (6) yields
e7
where is phrased as a draglike term in order to directly relate the power residual to changes in storm intensity. Dividing Eq. (7) through by gives
e8
The solution for υ at equilibrium in the absence of ventilation yields the traditional definition of the potential intensity (Bister and Emanuel 1998), given by
e9
Substitution of Eq. (9) into the first two terms on the RHS of Eq. (8) yields
e10
where . Dividing Eq. (10) through by gives
e11
where the tilde denotes nondimensionalization by (i.e., ). This may be written more simply as
e12
where
e13
is the (normalized) ventilation. Equation (12) is identical to Eq. (S14) from the TE12 supplement [note: the factor that appears in TE12’s Eq. (S14) cancels with its appearance in their definition of the normalized ventilation in TE12’s Eq. (S11) and so does not appear in Eqs. (12) and (13) here].

b. Analytical solutions for intensity dynamics

TE12 provided a graphical phase-space solution for as a joint function of current normalized intensity change and normalized ventilation (their Fig. 7), though they did not provide a corresponding analytical solution. Here we extend this derivation to analytical solutions for the predicted change in normalized intensity and associated extrema.

The change in normalized intensity may be written as
e14
where and are the current and final normalized intensities, respectively. The final intensity may be written in terms of changes in power as
e15
Using Eq. (12) to substitute for yields
e16
which gives
e17
Substituting Eq. (17) into Eq. (14) yields
e18
Equation (18) is the analytical solution for the predicted change in normalized intensity as a function of current normalized intensity and normalized ventilation; no explicit time scale is included in the theory, as noted in TE12. This analytical solution is displayed in Fig. 1 (color fill) and was verified to match the numerical solution to Eq. (12), both of which match Fig. 7 of TE12.
Fig. 1.
Fig. 1.

Analytical solution [Eq. (18)] for joint dependence of normalized intensity change on normalized intensity () and normalized ventilation . Purple line denotes equilibrium solution [Eq. (19)], including stable branch (solid), corresponding to the ventilation-dependent maximum potential intensity, and unstable branch (dashed), corresponding to the ventilation-dependent genesis threshold. Markers: × denotes the largest possible value of that can support a storm at equilibrium [Eq. (20)]; * denotes location of maximum normalized intensification rate [Eq. (21)] with value of annotated. For negative values of whose magnitude exceeds (i.e., storm lysis), is simply set equal to .

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

Equation (12) may be further manipulated to give two useful analytical results. First, as was done in TE10 [their Eq. (33)] and TE12, setting yields the equilibrium branches for the phase space given by the cubic equation
e19
This equilibrium solution is displayed in Fig. 1 (purple line). The upper, stable branch solution to Eq. (19) represents the ventilation-dependent maximum potential intensity. The lower, unstable branch represents the finite-amplitude, ventilation-dependent genesis threshold that separates intensifying storms above this branch from decaying storms below this branch. Differentiating Eq. (19) with respect to and setting equal to zero yields the solution for the largest possible value of that can support an equilibrium storm , given by
e20
Second, setting in Eq. (18), differentiating with respect to , and setting equal to zero yields the solution for the maximum possible normalized intensification rate , given by
e21
Each of the above analytical results is also displayed in Fig. 1. We note that peak entropy surplus, for which is maximized, occurs at a smaller normalized intensity of , where and . Finally, note that the combination of both of the above cases yields , for which the physically meaningful solutions are and ; the latter corresponds to {i.e., the canonical maximum potential intensity [Eq. (9)]}, which is the lone nonzero attractor in the ventilation-free system (Kieu 2015).

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 . This generalization will yield theoretical insight into the effects of such a cap on the dynamics of the system. In particular, we may directly test the assertion that the intensification of a tropical cyclone in the presence of a capped surface entropy flux wind speed is at odds with prevailing tropical cyclone theory. We note that there is no obvious physical context in which a capped surface entropy flux wind speed might occur, though it may be relevant to the case where decreases with increasing wind speed, which could occur at very high wind speeds within intense storms (Chen et al. 2013).

The only direct effect of the imposition of this wind speed cap is to set in the surface entropy flux term (when ) in Eq. (12); it also reduces the maximum potential intensity, which is incorporated into the solution in section 3b. This direct effect leads to a modified form of Eq. (12) given by
e22
This equation and associated solutions are identical to Eq. (12) for all . For , we obtain
e23
where , which represents the ratio of the capping wind speed to the potential intensity, is constant and is taken here to be the primary external parameter for the system.
Making use of Eqs. (14) and (15), the resulting change in normalized intensity is given by
e24
The equilibrium solution is evident directly from Eq. (24) by setting , which yields
e25
The result is a single stable equilibrium branch that represents the ventilation-dependent, capped-flux maximum potential intensity; there is no unstable lower branch. The absence of lower branch implies that there is no lower-bound threshold separating intensifying and decaying systems, except that this regime is only valid for . Following from this single-branch solution, direct inspection of Eq. (25) indicates that the largest possible value of that can support a nondecaying storm is given by . Equation (24) indicates that the normalized intensification rate decreases monotonically with increasing and , and thus the maximum normalized intensification rate is given by . However, the capped-flux regime only applies for , and thus and are never realized. Instead, occurs at ; for , the solution for is . Similarly, occurs at ; for , the solution for is .

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 . The imposition of a (normalized) capping surface entropy flux wind speed acts to shift the stable branch downward, thereby reducing the capped-flux potential intensity at all values of normalized ventilation. For sufficiently small capping wind speeds, the uncapped stable branch is completely replaced by the capped-flux stable branch solution. This solution also shifts to the left as the capping wind speed is further decreased, indicating a reduction in the maximum normalized ventilation at which a nondecaying storm may exist. This result matches the finding of Zhang and Emanuel (2016) that a storm embedded in sufficiently high wind shear may not intensify in the absence of the WISHE feedback; here, this is generalized to sufficiently high ventilation. The intensifying regime within the phase space disappears fully only in the limit of vanishing , at which point no storm can exist in the first place.

Fig. 2.
Fig. 2.

As in Fig. 1 for , but for the case of capped surface entropy fluxes [Eqs. (24) and (25)]. Subplots show the solution for . Gray line indicates normalized capping surface entropy flux wind speed . Circle denotes . Setting to be large fully recovers the result of Fig. 1.

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

b. Capped-flux potential intensity nondimensionalization

As noted above, capping the surface entropy flux wind speed also reduces the maximum potential intensity. The capped-flux potential intensity may be derived from Eq. (22) by setting the LHS and equal to zero, which gives
e26
Thus, for , Eq. (26) recovers . A similar result was derived by Zhang and Emanuel (2016) [their Eq. (10)].
Ideally we seek a solution of the same form as Eq. (22) but nondimensionalized by rather than , as the former is the relevant upper bound for the capped-flux system. Multiplication of Eq. (22) by the factor yields
e27
where the circumflex denotes nondimensionalization by in lieu of . Substitution for using Eq. (26) yields
e28
where
e29
Making use of Eqs. (14) and (15), the resulting change in normalized intensity is given by
e30
The equilibrium solution is directly evident from Eq. (30) by setting , which yields
e31
This form of the solutions given by Eqs. (28)(31) is again phrased in terms of the external parameter ; the capped-flux potential intensity is an internal parameter predicted by Eq. (26).

The above solution is displayed in Fig. 3 for the same range of values of external parameter as in Fig. 2. In contrast to Fig. 2, the upper equilibrium branch is now maximized at unity in all cases, as was the case with Fig. 1. Moreover, varying induces substantial variation in the magnitude of normalized intensity change ; peak values of are large for small and decrease with increasing , approaching the uncapped solution of Fig. 1 as as expected. However, the structure of the phase-space solution, in particular the equilibrium branches, remains nearly fixed as is varied, with the exception of modest changes toward the largest values of normalized ventilation that support an intensifying system.

Fig. 3.
Fig. 3.

As in Fig. 2, but with the system nondimensionalized by in lieu of [i.e., ; ; Eqs. (30) and (31)]. Subplots show the solution for the same range of values of as Fig. 2. Gray line indicates . Setting to be large fully recovers the result of Fig. 1.

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 (i.e., ), the potential intensity is unmodified , and thus the resulting solution is identical to that of section 3a. Second, for (i.e., ), in which the potential intensity is reduced, there are two possibilities: 1) {i.e., }, where the system does not yet feel the effect of the capping of the surface fluxes, and thus the solution is identical to the uncapped regime; and 2) {i.e., }, where the dynamics of υ (i.e., ) are affected by the flux capping. Note that the complete noncapped solution of Eq. (12) is recovered at all values of υ for (i.e., no capping wind speed).

The final regime is novel and is explored here. For this regime,
e32
As a result, . Thus, Eq. (28) reduces to
e33
Given that the form of Eq. (33) is identical to Eq. (23), the associated analytical solutions are qualitatively similar to the capped-flux system nondimensionalized by described above. The change in normalized intensity is given by
e34
Thus, the factor effectively acts to amplify the detrimental effect of ventilation on normalized intensity change. This simple theoretical result succinctly explains how a surface flux wind speed cap could inhibit intensification in high wind shear as discussed in the previous section. From Eq. (34), the equilibrium solution is given by
e35
which again yields a single stable equilibrium branch. Thus, Eq. (35) indicates that the largest possible value of that can support a nondecaying storm is identical in form to the previous section [Eq. (24)], given by , as . Equation (34) indicates that the normalized intensification rate decreases monotonically with increasing and , with the maximum normalized intensification rate given by . However, as in the previous section, the capped-flux regime of Eq. (35) only applies for , and thus and are never realized. Instead, occurs at ; for , the solution for is . Similarly, occurs at ; for , the solution for is . Here, and are related to [Eq. (20)] and [Eq. (21)], respectively, via the rescaling factor . As shown in Fig. 3, for strongly capped conditions (i.e., small ), the system approaches the solutions of Eqs. (34) and (35), with the maximum normalized intensity change approaching unity (i.e., an intensification from to ). However, this limit cannot in fact be reached since in the limit there are no surface fluxes at all, and thus .

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 in Eq. (34) could in principle be absorbed into to create a ventilation parameter nondimensionalized by in lieu of . This choice would align with the underlying physics of the normalized ventilation in which the denominator represents surface entropy fluxes. However, such an approach is not smoothly extended to the other regimes in which either or are unmodified by the capping wind speed .

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.

REFERENCES

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    • Search Google Scholar
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    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chen, S. S., W. Zhao, M. A. Donelan, and H. L. Tolman, 2013: Directional wind–wave coupling in fully coupled atmosphere–wave–ocean models: Results from CBLAST-Hurricane. J. Atmos. Sci., 70, 31983215, doi:10.1175/JAS-D-12-0157.1.

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1

Specifically, , where k is a proportionality constant between the background vertical wind shear and radial flow perturbations, and α represents the combination of several physical variables; the reader is referred to TE12 for details.

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