formulation of hypothesis of tropical cyclone

Climate Variability and Predictability Program

Tropical Cyclone Theory

by Kerry Emanuel and Adam Sobel

Predicting the response of tropical cyclones to changing climate should be aided by a better understanding of the factors that control their rate of generation, their intensification, and their demise, as well as related phenomena such as flooding and storm surges. Development and testing of comprehensive theories for these and other aspects of tropical cyclones is of fundamental scientific interest, and also is needed to help us predict how climate change might affect tropical cyclones and their effects on society.

Research Summary 

Research to date has established a few fundamental principles and, perhaps more importantly, has delineated important questions that must be answered to achieve a comprehensive understanding of the phenomenon. Early investigators (e.g. Riehl, 1950, Kleinschmidt, 1951) established that mature tropical cyclones are maintained against frictional dissipation by enthalpy (heat) transfer from the ocean to the atmosphere. In nature, most of this enthalpy transfer is accomplished by the evaporation of seawater. In this process, the latent heat of vaporization is supplied by the ocean, which acts as very large heat capacitor for the atmosphere on the time scales of tropical cyclones. (By contrast, the land surface has a vanishingly small effective heat capacity on tropical cyclone time scales, which explains why virtually all such storms dissipate when they move over land.) The mature tropical cyclone, approximated as a steady, circularly symmetric vortex, may be idealized as an ideal Carnot heat engine, as illustrated in Figure 1. As air spirals into the eyewall region (A to B in Figure 1), it undergoes a nearly isothermal expansion, acquiring enthalpy from the sea surface and does so. It then ascends in the eyewall, undergoing a nearly (moist) adiabatic expansion. Enthalpy acquired from the sea is either exported from the system or lost by infrared radiation to space, at the comparatively low temperature of the tropical tropopause region. In a closed cycle, the air would slowly subside and compress under the influence of radiative cooling, following the (moist) adiabatic temperature profile of the tropical environment. The thermodynamic efficiency of the cycle is proportional to the difference between the sea surface and tropopause temperatures.

A common misconception about tropical cyclones is that they are powered by the latent heat released as air ascends and expands in the eyewall. But this is an internal energy conversion and so does not bear on the overall energetics of the system. In fact, perfectly dry hurricanes have been simulated numerically (Mrowiec et al., 2011). In this case, the ascent and decent are dry adiabatic, and the radiative-convective equilibrium of the storm environment is likewise characterized by dry adiabatic lapse rates of temperature.

While the basic energy cycle helps explain the maintenance of tropical cyclone against frictional dissipation of wind, it does not by itself explain how such storms come into being in the first place, nor does it fully explain how storms, once initiated, intensify. It has been shown (e.g. Emanuel, 1989) that tropical cyclones cannot arise spontaneously but must be set off by some independent process or processes. Understanding those processes is an important endeavor known as the “genesis problem” in the science of tropical cyclones. There are many unresolved theoretical issues pertaining to tropical cyclones.

These include:

• What environmental and physical parameters determine the maximum achievable intensity of tropical cyclones? There are many related issues that must be resolved to answer this question. These include, but may not be limited to
• The physics of air-sea enthalpy and momentum transport at high wind speeds, including such factors as waves and sea spray (Fairall et al., 1994, Edson et al., 1996, Andreas and Emanuel, 2001)
• The existence and magnitude of supergradient winds in the boundary layer (Smith et al., 2008, Bryan and Rotunno, 2009a)
• Horizontal mixing by atmospheric eddies (Bryan and Rotunno, 2009b)
• Radial structure of the “outflow temperature” (Emanuel and Rotunno 2011)
• What environmental factors control the actual (as opposed to potential) intensity of tropical cyclones? Answering this question involves such issues as
• Response of the upper ocean to tropical cyclones; especially, cooling of the sea surface by vertical mixing in the ocean (Khain and Ginis, 1991)
• Interaction of tropical cyclones with environmental winds, which may serve to import low entropy air into the storm core (Tang and Emanuel, 2010)
• The time it takes for storms to intensify, as they are more likely to reach their maximum potential intensities if they have time to do so before reaching land or colder ocean surfaces.
• How do Tropical Cyclones, once initiated, intensify? (Emanuel, 1997, Smith and Montgomery, 2009)
• How do tropical cyclones form? This is one of the great, largely unsolved problems of tropical meteorology. While empirical necessary conditions for tropical cyclone formation have been known for many decades (e.g., Palmen 1948; Gray 1979), a fundamental understanding of genesis remains elusive.
• What controls the sizes of tropical cyclones? While there have been recent advances in empirical knowledge of storm size distributions (e.g. Chavas and Emanuel, 2010), there is almost no theoretical understanding of storm size.
• What determines the level of tropical cyclone activity in a given climate state?

This is a subject of intense theoretical as well as modeling-based research, and involves many of the issues described above. The term “tropical cyclone activity” includes the frequency and intensity of events as well as their characteristic tracks, regions of genesis, and horizontal dimensions. These statistics of the tropical cyclone distribution are known to vary with natural variations of the large-scale climate such as ENSO (e.g., Camargo et al. 2010, and references therein), and it is natural to expect that they will vary with anthropogenic global warming as well. However, our understanding of how climate variations control TC statistics is based largely on empirical methods (which are limited by the observational record and contain no true analog for global warming) and now, to an increasing extent, global high-resolution numerical models (e.g., Knutson et al. 2010). Theory is so far silent on several of these important questions; there is no existing theory, for example, which predicts the average number of tropical cyclones thaform in a year under a given climate.

Implications

Tropical cyclones are, by far, the major source of insured losses by natural catastrophes worldwide, and are a leading cause of damage and mortality from natural phenomena. Even small changes in their levels of activity owing to climate change may have important societal consequences. Successful prediction of the response of tropical cyclone activity to climate change is predicated not just on better numerical simulations but on much improved theoretical understanding of these storms.

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• 1. Introduction
• 2. E-PI derivation for the point where ∂υ/∂z = 0
• 3. E-PI’s key relationship
• a. Constraints on ∂T/∂r from the definition of saturation entropy
• b. Thermal wind constraints on ∂T/∂r
• 5. Constraints from the equations of motion
• 6. Implications for the boundary layer closure in E-PI
• 7. Discussion and conclusions
• Acknowledgments.
• Data availability statement.
• How Does the Gradient Wind Change over z?
• Deriving an Alternative for E-PI’s Eq. (13)

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Bryan , G. H. , and R. Rotunno , 2009b : The maximum intensity of tropical cyclones in axisymmetric numerical model simulations . Mon. Wea. Rev. , 137 , 1770 – 1789 , https://doi.org/10.1175/2008MWR2709.1 .

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Hazelton , A. T. , and R. E. Hart , 2013 : Hurricane eyewall slope as determined from airborne radar reflectivity data: Composites and case studies . Wea. Forecasting , 28 , 368 – 386 , https://doi.org/10.1175/WAF-D-12-00037.1 .

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Kowaleski , A. M. , and J. L. Evans , 2016 : A reformulation of tropical cyclone potential intensity theory incorporating energy production along a radial trajectory . Mon. Wea. Rev. , 144 , 3569 – 3578 , https://doi.org/10.1175/MWR-D-15-0383.1 .

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Makarieva , A. M. , and A. V. Nefiodov , 2022 : Alternative expression for the maximum potential intensity of tropical cyclones. arXiv, 2101.06500v4, https://doi.org/10.48550/arXiv.2101.06500 .

Makarieva , A. M. , and Coauthors , 2023 : Water lifting and outflow gain of kinetic energy in tropical cyclones . J. Atmos. Sci. , https://doi.org/10.1175/JAS-D-21-0172.1 , in press.

Montgomery , M. T. , M. M. Bell , S. D. Aberson , and M. L. Black , 2006 : Hurricane Isabel (2003): New insights into the physics of intense storms. Part I: Mean vortex structure and maximum intensity estimates . Bull. Amer. Meteor. Soc. , 87 , 1335 – 1348 , https://doi.org/10.1175/BAMS-87-10-1335 .

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Peng , K. , R. Rotunno , and G. H. Bryan , 2018 : Evaluation of a time-dependent model for the intensification of tropical cyclones . J. Atmos. Sci. , 75 , 2125 – 2138 , https://doi.org/10.1175/JAS-D-17-0382.1 .

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Puy , A. , P. Beneventano , S. A. Levin , S. L. Piano , T. Portaluri , and A. Saltelli , 2022 : Models with higher effective dimensions tend to produce more uncertain estimates . Sci. Adv. , 8 , eabn9450 , https://doi.org/10.1126/sciadv.abn9450 .

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Stern , D. P. , J. R. Brisbois , and D. S. Nolan , 2014 : An expanded dataset of hurricane eyewall sizes and slopes . J. Atmos. Sci. , 71 , 2747 – 2762 , https://doi.org/10.1175/JAS-D-13-0302.1 .

Tao , D. , K. Emanuel , F. Zhang , R. Rotunno , M. M. Bell , and R. G. Nystrom , 2019 : Evaluation of the assumptions in the steady-state tropical cyclone self-stratified outflow using three-dimensional convection-allowing simulations . J. Atmos. Sci. , 76 , 2995 – 3009 , https://doi.org/10.1175/JAS-D-19-0033.1 .

Tao , D. , M. Bell , R. Rotunno , and P. J. van Leeuwen , 2020 : Why do the maximum intensities in modeled tropical cyclones vary under the same environmental conditions? Geophys. Res. Lett. , 47 , e2019GL085980 , https://doi.org/10.1029/2019GL085980 .

Parameters ε ≡ ( T − T o )/ T vs 1/(1 + ζ ) as dependent on temperature T ; ε curves correspond to different outflow temperatures T o ; 1/(1 + ζ ) curves correspond to p d values of 800 and 900 hPa; see Eq. (B10) .

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A Critical Analysis of the Assumptions Underlying the Formulation of Maximum Potential Intensity for Tropical Cyclones

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Emanuel’s concept of maximum potential intensity (E-PI) estimates the maximum velocity of tropical cyclones from environmental parameters assuming thermal wind (gradient-wind and hydrostatic balances) and slantwise neutrality in the free troposphere. E-PI’s key equation relates proportionally the radial gradients of saturated moist entropy and angular momentum. Here the E-PI derivation is reconsidered to show that the thermal wind and slantwise neutrality imply zero radial gradients of saturation entropy and angular momentum at an altitude where, for a given radius, the tangential wind has a maximum. It is further shown that, while E-PI’s key equation requires that, at the point of maximum tangential wind, the air temperature must increase toward the storm center, the thermal wind equation dictates the opposite. From the analysis of the equations of motion at the altitude of maximum tangential wind in the free troposphere, it is concluded that here the airflow must be supergradient. This implies that the supergradiency factor (a measure of the gradient-wind imbalance) must change in the free troposphere as the airflow tends to restore the balance. It is shown that such a change modifies the derivative of saturation entropy over angular momentum, which cannot therefore remain constant in the free troposphere as E-PI requires. The implications of these findings for the internal coherence of E-PI, including its boundary layer closure, are discussed.

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy ( www.ametsoc.org/PUBSReuseLicenses ).

Tropical storms threaten human lives and livelihoods. Numerical models can simulate a wide range of storm intensities under the same environmental conditions (e.g., Tao et al. 2020 ). Thus, it is desirable to have a reliable theoretical framework that would, from the first principles, confine model outputs to the domain of reality ( Emanuel 2020 ). The theoretical formulation for maximum potential intensity (E-PI) of tropical cyclones by Emanuel (1986) has been long considered as an approximate upper limit on storm intensity [see discussions by Garner (2015) , Kieu and Moon (2016) , and Kowaleski and Evans (2016) ]. Studies have shown that the maximum wind (observed or modeled) can be larger than E-PI due to supergradient wind (“superintensity”) (e.g., Persing and Montgomery 2003 ; Montgomery et al. 2006 ; Bryan and Rotunno 2009a ; Rousseau-Rizzi and Emanuel 2019 ; Li et al. 2020 ). Here we reconsider the assumptions behind E-PI to show that they are mutually incompatible at the point of maximum wind.

The E-PI formulation is based on the thermal wind equation and the assumption of slantwise neutrality in the free troposphere. In section 2 we repeat the E-PI derivation following Emanuel (1986) but focusing on the altitude where tangential velocity has a local maximum ∂ υ /∂ z = 0. We show that here E-PI predicts zero radial gradients of saturation entropy s * and angular momentum thus not permitting nontrivial solutions and being inapplicable for the assessment of maximum winds.

For readers immediately interested in the underlying physics, here is a brief explanation. The gradient-wind balance consists in the equality of the centripetal and centrifugal forces: the radial pressure gradient per unit density and the squared tangential velocity divided by radius. Where ∂ υ /∂ z = 0, the latter force is invariant over z . For the thermal wind equation to apply, the gradient wind (determined by the radial pressure gradient per unit air density) must also be constant over z . In hydrostatic equilibrium, this is the case when the radial and vertical gradients of temperature T over pressure p are equal (see appendix A ). When ∂ s * / ∂ z = ∂ s * / ∂ r = 0 , this condition is fulfilled: the temperature gradients in both directions are moist adiabatic. In real cyclones, the radial pressure gradient diminishes with height changing its sign in the midtroposphere. The thermal wind equation at the point ∂ υ /∂ z = 0 cannot hold. Another perspective on the same problem is that E-PI constrains the slope of angular momentum surfaces, and this predicted slope is never zero—although it must be so where ∂ υ /∂ z = 0 (see section 6 ).

In section 3 we discuss why the incompatibility between E-PI’s assumptions is not explicit in the resulting E-PI formula. In section 4 we show how this incompatibility can be explicated by combining the E-PI formula with the definition of saturated moist entropy. This reveals that the E-PI formula and the thermal wind equation from which it derives, predict the opposite signs of the radial temperature gradient at the point of maximum tangential wind. In section 5 we discuss additional dynamic constraints on E-PI from the equations of motion. In section 6 we discuss the implications of our findings for the boundary layer closure in E-PI. In view of the obtained results, the concluding section 7 discusses the general coherence of E-PI and some issues with its verification by numerical modeling.

2. E-PI derivation for the point where ∂ υ / ∂ z = 0

Whenever ∂ υ /∂ z = 0, we have ∂ M /∂ z = 0 and (∂ M /∂ p ) r = 0 and, by consequence from Eq. (8) , ( ∂ s * / ∂ r ) p = 0 . On the other hand, since s * = s * ( M ) and ( ∂ s * / ∂ p ) r = ( d s * / d M ) ( ∂ M / ∂ p ) r , we also have ( ∂ s * / ∂ p ) r = 0 (excluding the unrealistic case d s * / d M = 0 ). But since ( ∂ s * / ∂ p ) r = 0 and ( ∂ s * / ∂ r ) p = 0 , this means that whenever ∂ υ /∂ z = 0, the radial gradient of saturation entropy is zero: ∂ s * / ∂ r = 0 . Furthermore, since ∂ s * / ∂ r = ( d s * / d M ) ∂ M / ∂ r , the radial gradient of angular momentum is also zero: ∂ M /∂ r = 0. These conclusions do not depend on the value of b .

Our conclusion so far is that the thermal wind equation and the assumption of slantwise neutrality are incompatible with ∂ υ /∂ z = 0.

We will now see why this incompatibility is not explicit in the resulting E-PI formula. We put (∂ b /∂ p ) r = 0 in Eq. (8) . With b = 1, the four equations below correspond to Emanuel’s (1986) Eqs. (10)–(13).

These derivations, of Bryan and Rotunno (2009a) , Makarieva et al. (2023) , and Eqs. (10) – (13) with a constant b ≠ 1, assume that in the free troposphere the air motion conserves not only the angular momentum, but also the supergradiency factor b ≠ 1. Such motion, while mathematically possible, is not physically plausible: in the real free troposphere the flow will tend to restore the gradient-wind balance, i.e., b ≠ 1 will change to b ≃ 1 (with a minor deviation from unity determined by how small the turbulent friction is). If, as it enters the free troposphere, the airflow is supergradient with b > 1, then, as it begins to relax to gradient balance, (∂ b /∂ p ) r > 0 in Eq. (8) . The absolute magnitude of d s * / d M retrieved from Eq. (10) is then smaller than it is when (∂ b /∂ p ) r = 0. This indicates that | d s * / d M | should increase in the upper troposphere where the air reaches gradient-wind balance [ b ≃ 1 and (∂ b /∂ p ) r ≃ 0].

Focusing on the point where ∂ υ /∂ z = 0, we notice that E-PI’s key equation was obtained by dividing Eq. (8) by (∂ M /∂ r ) p = 0, integrating the resulting equation along M surface, and multiplying it again by ∂ M /∂ r = 0. In the resulting formula, after this dividing and multiplying by zero, the inapplicability of E-PI to the point where ∂ υ /∂ z = 0 became implicit. However, as we show in the next section, it can be explicated at the point of maximum tangential wind.

4. Radial temperature gradient at the point of maximum tangential wind

We will now show that, at the point of maximum tangential wind, where ∂ υ /∂ r = ∂ υ /∂ z = 0, E-PI’s key equation and the thermal wind equation, from which the former is derived, predict the opposite signs for the radial temperature gradient.

The maximum Carnot efficiency estimated from temperatures T o and T = T b observed, respectively, in the outflow and at the top of the boundary layer, is ε = 0.35 ( DeMaria and Kaplan 1994 ). Assuming that T b does not usually exceed 303 K (30°C), the minimum value of (1 + ζ ) −1 ≃ 0.5 is larger than ε . It corresponds to the largest γ d * ≃ 0.05 for T b = 303 K and p d ≃ 800 hPa. The partial pressure p υ * of saturated water vapor and, hence, γ d * depend exponentially on air temperature. The realistic temperatures at the top of the boundary layer are commonly significantly lower than 303 K.

Thus, under observed atmospheric conditions [ ε (1 + ζ )] −1 > 1 ( Fig. 1 ). This means that for the E-PI cyclone to exist, i.e., for ∂ p /∂ r > 0, the air temperature must grow in the direction of the cyclone center, i.e., ∂ T /∂ r < 0 where ∂ υ /∂ r = 0. We emphasize that, to be valid, E-PI requires a specific value of ∂ T /∂ r < 0 as determined by Eq. (18) . At observed temperatures, E-PI requires C ≃ 2 [i.e., (∂ T /∂ p ) z ≃ −Γ]. Notably, due to the change of the sign of the last term in Eq. (19) at high temperatures, see Fig. 1 , E-PI requires that at high temperatures the air temperature at the point of maximum tangential wind should decline in the direction of the storm center, i.e., C < 1 .

Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JAS-D-22-0144.1

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This result can be directly derived from the thermal wind equation, Eq. (6) of Emanuel (1986) and Eq. (5) of Emanuel and Rotunno (2011) ; see also our Eq. (6) . It says that where the balanced wind is maximum over z , (∂ M /∂ p ) r = 0, we have (∂ α /∂ r ) p = 0 and, hence, (∂ T /∂ r ) p = 0. In the boundary layer of tropical cyclones, the isobars rise outward from the center, (∂ z /∂ r ) p > 0. With ∂ T /∂ z < 0, the coincidence of isobars and isotherms means that ∂ T /∂ r > 0.

For the condition ∂ T /∂ r < 0 to hold, it follows from Eq. (22) that the supergradiency factor b must increase with altitude at the point of maximum tangential wind, ∂ln  b /∂ z > −∂ln  T /∂ z > 0. We will now show that b = 1 is incompatible with ∂ b /∂ z ≠ 0 where ∂ υ /∂ z = 0. It is not possible to retain the gradient-wind balance assumption locally but to relax it in the vicinity of this point.

For ∂ υ /∂ z = 0, we have from Eq. (24b) that u = 0. If b = 1 in Eq. (3) , the sum of the last two terms in Eq. (24a) is zero. For u = 0 and w ≠ 0 (the eyewall), 3 this means that ∂ u /∂ z = 0. With u = 0 and ∂ u /∂ z = 0, the first three terms in the right-hand part of Eq. (25) are zero.

The radial velocity u changes its sign at the point of maximum tangential wind, where u = 0. Below this point, there is convergence and u < 0, while above this point there is divergence and u > 0. Usually, the horizontal level that separates u < 0 and u > 0 is close to the top of the boundary layer; see, e.g., Bryan and Rotunno’s (2009a) Fig. 11 for modeling and Montgomery et al.’s (2006) Fig. 4b for real cyclones. From the conditions that ∂ u /∂ z = 0 at the point where ∂ υ /∂ z = 0 and ∂ u /∂ z ≥ 0 in the vicinity of this point, 4 it follows that the second derivative of u with respect to z is zero at the point where ∂ υ /∂ z = 0. Then the fourth term in the right-hand part of Eq. (25) is zero as well. This means that ∂ b /∂ z = 0, if b = 1 where ∂ υ /∂ z = 0. This shows that it is generally not possible to specify b and ∂ b /∂ z independently. 5

In the eyewall with ∂ u /∂ z > 0 and w > 0, it follows from Eqs. (24a) and (24b) that b > 1 for ∂ υ /∂ z = 0. In other words, those tropical cyclones that have their maximum wind in the free troposphere must be supergradient [cf. Eq. (9) ]. The conventional balanced E-PI, which assumes b = 1 in the free troposphere, has no solutions under observed atmospheric conditions. With b ≠ 1 unknown, E-PI is not a closed theory.

We are now in a position to discuss where ∂ υ /∂ z = 0 is realized in real cyclones and in models. Surprisingly, despite all the research emphasis on maximum potential intensity , the question of where this maximum is located along the vertical axis does not appear to have received consistent attention: observational studies of vertical υ profiles exclude the boundary layer (see below). For case studies, Montgomery et al. (2006 , their Fig. 4a) reported that, for Hurricane Isabel (2003), the mean tangential wind in the eyewall (40 ≤ r ≤ 50 km) has a maximum at a height of about 1 km, where it is approximately 50% greater than its surface value of approximately 50 m s −1 . Hurricanes Ivan (2004), Wilma (2005), Frances (2004), Helene (2006), and Dennis (2005), as shown, respectively, in Figs. 1c and 7b–d of Stern et al. (2014) and Fig. 5a of Stern and Nolan (2009) , display the same feature: for r ≳ 40 km , the slopes of υ contours change sign, thus indicating a maximum of υ , at or below 1 km. Peng et al.’s (2018) Figs. 4 and 5 likewise show a maximum of tangential wind at ∼0.5 km altitude for their simulated cyclones. Thus, the increase of tangential wind with altitude with a maximum near the top of the boundary layer in the eyewall appears to be a common feature in real storms as well as in models.

In E-PI, Emanuel’s (1986) Fig. 1 presents a scheme of the boundary layer with constant s * surfaces that are vertical and constant M surfaces that are not vertical, but become approximately so at the boundary layer top. This scheme is consistent with the assumption Bryan and Rotunno (2009a , p. 3045) assigned to E-PI, that the maximum gradient wind “is located at the top of the boundary layer” where “viscous terms become negligible.” Emanuel and Rotunno (2011 , p. 2239) explained that for the boundary layer closure in E-PI it is sufficient that “entropy is well mixed along angular momentum surfaces, which are approximately vertical in the boundary layer,” and vertical is how Peng et al. (2018) show these surfaces in their Fig. 10b for E-PI. Smith et al. (2008 , p. 553) also interpreted E-PI’s boundary layer closure as presuming ∂ υ /∂ z = 0 within the boundary layer. Approximately vertical υ contours near the radius of maximum wind can be observed in real storms as well [see, e.g., Stern et al. (2014) , their Figs. 1a and 1b for Hurricane Ivan (2004)].

In Bryan and Rotunno’s (2009a) control simulation, tangential velocity in the eyewall increases with altitude within the lower 1 km (see their Fig. 4b). According to Bryan and Rotunno (2009a , p. 3050), the assumption that the maximum tangential velocity is achieved at the top of the boundary layer “is needed to match the free-atmosphere component to the boundary layer closure in E-PI.” In some discrepancy with this interpretation, Stern and Nolan (2011 , their Figs. 5a,c) indicated that E-PI, rather, presumes that the maximum tangential wind is located at the surface z = 0, where ∂ υ /∂ z < 0, and monotonously declines with height. 6 Likewise, according to Rousseau-Rizzi and Emanuel (2019) , E-PI’s maximum tangential wind at the surface exceeds the maximum tangential wind at the boundary layer top. This provides a complementary perspective on the discussed incompatibility between E-PI’s assumptions at ∂ υ /∂ z = 0. The thermal wind equation and the assumption of slantwise neutrality constrain the slope of the angular momentum surfaces [see Stern and Nolan 2009 , their Eq. (A.14)]. This predicted slope is never vertical (unless r = 0), although it must be so where ∂ υ /∂ z = 0.

If the M and s * surfaces are vertical within the boundary layer, and then on the top of the boundary layer their slope abruptly changes to the one constrained by E-PI’s Eq. (12) for the free troposphere, then the radial gradients of M , s * , T , and p will all have infinite derivatives over z on the top of the boundary layer where this discontinuity occurs. According to Emanuel and Rotunno (2011) , E-PI’s boundary layer closure requires s * to be well mixed along the approximately vertical M surfaces. The condition ∂ s * / ∂ z = 0 is generally incompatible with ∂ 2 s * / ( ∂ z ∂ r ) ≠ 0 resulting from the discontinuity on the boundary layer top ( ∂ s * / ∂ z can be zero only at a certain radius where it must change sign over r ). It is therefore pertinent to check to which degree the approximation of verticality is essential for E-PI’s boundary layer closure.

E-PI’s boundary layer closure constrains d s * / d M = ( ∂ s * / ∂ M ) z at the top of the boundary layer as the ratio of the vertical fluxes F s * and F M of s * and M at the sea surface. Originally, Emanuel (1986 , p. 593) applied his Eq. (27), valid “for any conservative variable c assumed to be well-mixed in the vertical within a turbulent boundary layer” to angular momentum M , but noted at the same time that M “may not be well mixed in the vertical.” 7

In the general case of ∂ M / ∂ z ≠ 0 and ∂ s * / ∂ z ≠ 0 , E-PI’s boundary layer closure requires s * = s * ( M ) within the boundary layer. Writing Eq. (26) for c = s * and using ∂ s * / ∂ r = ( d s * / d M ) ∂ M / ∂ r and ∂ s * / ∂ z = ( d s * / d M ) ∂ M / ∂ z , and Eq. (28) , we obtain d s * / d M = ( ∂ F s * / ∂ z ) / ( ∂ F M / ∂ z ) . If d s * / d M is constant and if both fluxes become zero at the top of the boundary layer, this can be integrated over z to yield d s * / d M = F s * / F M . But s * = s * ( M ) within the boundary layer cannot be justified due to turbulence.

Emanuel and Rotunno (2011 , p. 2239) referred to the study of Bryan and Rotunno (2009a) as demonstrating that E-PI’s boundary closure “is well satisfied in axisymmetric numerical simulations.” However, in the control simulation of Bryan and Rotunno (2009a , p. 3049), E-PI’s boundary layer closure at the radius of maximum wind is violated by 50%: the diagnosed ratio of surface fluxes is 1.5-fold greater than the diagnosed ( ∂ s * / ∂ M ) z at the top of the boundary layer, as shown in Bryan and Rotunno’s (2009a) Fig. 6. This discrepancy is smaller in their Fig. 7, which presents the simulations of Bryan and Rotunno (2009b) . But those simulations were made with a different parameter l υ that controls vertical turbulence effects [ l υ = 100 m in the control simulation of Bryan and Rotunno (2009a , their Fig. 6) and l υ = 200 m for simulations of Bryan and Rotunno (2009b) shown in Bryan and Rotunno’s (2009a) Fig. 7].

Importantly, according to Bryan and Rotunno (2009b , see their Fig. 2), the value of l υ does not influence the maximum wind speed. At the same time, as the comparison of Bryan and Rotunno’s (2009a) Figs. 6 and 7 suggests, parameter l υ is instrumental in bringing E-PI’s boundary layer closure in agreement with the simulations. If there exist model parameters that control whether E-PI’s boundary layer closure is satisfied, and if such parameters make no impact on the maximum intensity, the inference is that the maximum intensity may not be as profoundly dependent on local surface fluxes as E-PI presumes. This requires further clarifications.

We applied E-PI’s assumptions to the altitude of maximum tangential wind (∂ υ /∂ z = 0), which, according to observations and numerical models, is located near the top of the boundary layer. We showed that here E-PI’s assumptions are mutually incompatible and only allow for a trivial solution ∂ s * / ∂ r = 0 and ∂ M /∂ r = 0. We also applied E-PI’s assumptions to the point of maximum tangential wind (∂ υ /∂ z = ∂ υ /∂ r = 0) and showed that here their mutual incompatibility results in contrasting predictions concerning the radial temperature gradient. The thermal wind equation requires it to be positive, while E-PI’s key equation constrains it to be negative and dependent on the outflow temperature.

E-PI is based on merging the free troposphere constraints with the boundary layer constraints. The incompatibility of its assumptions pertains to the altitude of maximum tangential wind located on the border between the two atmospheric layers, and has implications for both. We have shown that at the altitude of maximum tangential wind the flow must be supergradient and that its relaxation to the gradient-wind balance in the free troposphere disturbs the constancy of d s * / d M required by E-PI. In the boundary layer, the verticality of M surfaces assumed in E-PI from the sea surface up to the boundary layer top, is not compatible with the nonverticality of M surfaces required by E-PI in the free troposphere. This disturbs the relationship between d s * / d M on the boundary layer top and the ratio of the surface fluxes of s * and M that is required for E-PI’s boundary layer closure.

Without addressing these theoretical issues, continued efforts to verify E-PI, or its elements, with numerical simulations may not be conclusive regarding the general validity of E-PI. Increasing model complexity without a matching increase in the quality of its independent constraints leads to fuzzier conclusions ( Puy et al. 2022 ). In such a situation, the results of numerical simulations can be misleading. We discuss one example below.

From this one could conclude that E-PI’s Eq. (12) could be valid if not at the point of maximum tangential wind but at least at a certain altitude where the gradient-wind balance (approximately) holds. But there is an additional caveat. Tao et al. (2019 , p. 2999) correctly noted, see Eq. (11) above, that in E-PI d s * / d M should be constant on M surfaces because of the assumed congruence of s * and M surfaces. Thus, Tao et al. (2019 , p. 2999 and their Fig. 4) diagnosed d s * / d M not at the same point where they diagnosed the tangential wind (at the boundary layer top), but in the outflow region in the upper troposphere. However, their own Figs. 3b, 3d, and 3f make it clear that s * and M surfaces are not congruent over much of the eyewall (between approximately 2 and 9 km). This means that the values of d s * / d M in the boundary layer and in the outflow region cannot be assumed equal, as d s * / d M varies where s * ≠ s * ( M ) .

While Tao et al. (2019) did not analyze how d s * / d M varied along the M surfaces, other studies indicate that such variation can be substantial. Figure 5b of Peng et al. (2018) shows simultaneously s * and M contours and readily allows for the estimation of d s * / d M in their modeled steady-state vortex. In the boundary layer at the radius r = 30 km of maximum wind there are three M contour intervals in one s * contour interval, while in the outflow region at r = 80 km there are only two. This indicates a 1.5-fold increase of the absolute value of d s * / d M . This increase displays a tendency to continue at larger radii in the outflow [not shown in Peng et al.’s (2018) Fig. 5b]. This is consistent with the decline in | d s * / d M | for supergradient wind discussed in section 3 . If a similar pattern holds for the simulations of Tao et al. (2019) , then their use of d s * / d M from the outflow region would lead to an overestimate of υ 2 as diagnosed from Eq. (12) by a factor of 1.5 or more (cf. Peng et al. 2018 ).

In summary, we are not aware of any studies, either observational or modeling, where the validity of E-PI’s Eqs. (12) and (13) would be demonstrated together with the validity of their underlying assumptions. Our alternative Eq. (17) suggests that if a constraint on C (the degree of adiabaticity of the radial temperature gradient) at the point of maximum wind is found, one could dispense with the consideration of the upper troposphere—as E-PI dispenses with the consideration of boundary layer dynamics when constraining C from the free troposphere considerations alone [see Eq. (18) ]. Considering the boundary layer dynamics, Makarieva and Nefiodov (2022) suggested that ∂ T /∂ r = 0 and C = 1 at the radius of maximum wind is a plausible assumption, which accuracy could be further investigated.

However, from our perspective, the main, and fundamental, problems of E-PI [and of any other local approach, including the alternative Eq. (17) ] pertain to the boundary layer closure. Some were discussed here, but see also Makarieva and Nefiodov (2022) . Storm intensity is an integral property of the entire storm’s energetics, whereby the energy released over a large area is concentrated in the eyewall to generate maximum wind. It cannot be a local function of the highly variable heat input at the radius of maximum wind (even if one could tune a model to suggest otherwise). We argue for a principally different approach to storm dynamics.

Makarieva et al. (2023) estimated that the effect of condensate loading on E-PI’s formulation is minor.

Equations (18) and (19) clarify why hypercanes cannot exist. With ε (1 + ζ ) → 1, C → 1 and ∂ T /∂ r → 0. The radial pressure gradient in Eq. (19) is then undefined rather than infinite (cf. Emanuel 1988 ), see also Makarieva and Nefiodov (2022 , their appendix B) .

Bryan and Rotunno (2009a , p. 3054) noted that in numerical simulations the point of maximum tangential wind often coincides with the point of maximum vertical wind and that “numerical simulations and observations often show that u ≈ 0 at the location of maximum tangential velocity.”

In mathematical analysis, this point is called a stationary point of inflection , or saddle point . It is the point on a curve at which the curvature changes sign.

Smith et al. (2008) brought up a related argument. They stated that, with b = 1 at the top of boundary layer, E-PI implicitly assumed gradient-wind balance within the boundary layer. Emanuel and Rotunno (2011 , p. 2239) replied that the boundary layer closure in E-PI did not require such an explicit assumption. Smith et al. (2008 , p. 553) were correct for their particular model of the boundary layer, which assumed ∂ u /∂ z = ∂ υ /∂ z = 0. In this case Eq. (25) yields ∂ b /∂ z = 0 from b = 1.

Stern and Nolan (2011) did not verify this pattern from observations, as they confined their consideration to above 2 km. Subsequent studies retained this limitation ( Hazelton and Hart 2013 ; Stern et al. 2014 ); the recent study of Fischer et al. (2022) does not show the lower 2 km in their Fig. 8 for the vertical profiles of tangential wind—despite the data being available down to the lowest 500 m.

For example, in Bryan and Rotunno’s (2009a) control simulation designed to check the E-PI assumptions, M contours near the surface are approximately horizontal (see their Fig. 4).

The authors are grateful to three reviewers for their useful comments. Work of A. M. Makarieva is partially funded by the Federal Ministry of Education and Research (BMBF) and the Free State of Bavaria under the Excellence Strategy of the Federal Government and the Länder, as well as by the Technical University of Munich–Institute for Advanced Study. The authors thank Václav Vacek, Jan Pokorný, and Milan Vlach for stimulating discussions and support.

There were no raw data utilized in this study.

How Does the Gradient Wind Change over z ?

With ∂ υ /∂ z = 0, the supergradiency factor b remains constant over z if the horizontal and vertical gradients of temperature over pressure are equal ( K = 1). If the atmosphere is horizontally isothermal ( K = 0), then, with a moist adiabatic lapse rate −∂ T /∂ z ≃ 5 K km −1 , the relative increase in b will be under 2% over 1 km. If the horizontal temperature lapse rate is minus moist adiabatic ( K ≃ − 1 , C ≃ 2 ), as E-PI approximately requires (see section 4a ), then b will increase by no more than 4% over 1 km.

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