1. Introduction

The term “hurricane superintensity” has been used to denote that the maximum wind speed of a tropical cyclone exceeds the value predicted by Emanuel’s theory for potential intensity [hereafter referred to as E-PI; see Emanuel 1986 (hereafter E86) and Emanuel 1995]. Superintensity was first detected by Persing and Montgomery (2003) in the axisymmetric model of Rotunno and Emanuel (1987) at a high model resolution. They explained the simulated superintensity with the so-called “turbocharger” mechanism by which high-entropy air of the eye is injected into the eyewall. Bryan and Rotunno (2009b) found this mechanism to be rather unimportant for explaining the model superintensity. Instead, they emphasize the importance of horizontal momentum exchange by demonstrating an increase of superintensity with a decreasing momentum exchange coefficient (Bryan and Rotunno 2009c). Furthermore, Bryan and Rotunno (2009a, hereafter BR09) proposed an analytical model in which gradient wind imbalance accounts for hurricane superintensity.

In this study we suggest another possible cause to explain superintensity occurring in high-resolution tropical cyclone simulations. The E-PI theory is based on the assumption of zero slantwise convective available potential energy (SCAPE).¹ In this case isolines of angular momentum and entropy coincide as illustrated in Fig. 1a.² However, numerical simulations of tropical cyclones (e.g., Frisius and Hasselbeck 2009) and observations indicate that SCAPE exists in tropical cyclones (e.g., Sheets 1969; Frank 1977; Bogner et al. 2000; Eastin et al. 2005). Indeed, SCAPE could have an impact on increasing the intensity as illustrated in Fig. 1b: Given that SCAPE exists in the boundary layer, the air flowing into the eyewall has a larger entropy than the ambient air. As a consequence, the radial temperature gradient may become very large, leading to a larger intensity, whereas E-PI predicts a smoother radial temperature gradient at the outer boundary of the eyewall. This is due to the assumption that the specific saturation entropy takes the values of the boundary layer beneath. However, there is no evidence for this assumption since observations and numerical simulations do not support the conclusion that SCAPE vanishes in tropical cyclones. Abandoning the condition of zero SCAPE in potential intensity theory (hereafter referred to as PI theory) introduces a new parameter that represents the radial temperature gradient in the eyewall which should mainly be affected by radial turbulent heat transfer. Therefore, a modified PI approach based on SCAPE is consistent with the findings of Bryan and Rotunno (2009c).

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Sketch of entropy (solid lines) and angular momentum (dotted lines) in a tropical cyclone as they would appear in (a) the E-PI model and (b) an extended-PI model in which SCAPE exists.

Citation: Journal of the Atmospheric Sciences 69, 2; 10.1175/JAS-D-11-064.1

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We propose a modified PI model by extending E-PI so that SCAPE arises in the boundary layer outside of the eyewall and with an eye included in the center. The model is similar to that of E86 except for additional modifications to clarify the roles of SCAPE and the existence of an eye for hurricane superintensity. Section 2 closely reviews E-PI theory. In section 3, an extended-PI model is derived. Section 4 presents analytical results of the extended-PI model and numerical results of TC simulations by the axisymmetric cloud model Hurricane Model (HURMOD) to verify the concept of the extended-PI approach. Section 5 discusses other theories of potential intensity and their relation to the present extended-PI model. Section 6 summarizes the results of this study.

2. E-PI

In this section, E-PI theory is rederived in a concise way to form the basis for the extended-PI model. E86 neglected the influence of the Coriolis force several times in his derivation. For reasons of consistency, we disregard the Coriolis force from the outset. This is possible since the cyclostrophic balance already explains the essential cyclone structure and we do not consider the cyclogenesis process here, which cannot take place in the absence of the Coriolis force.

E-PI theory is based on a steady-state model for a balanced tropical cyclone. Here, we use potential radius R instead of physical radius r as the radial coordinate. The basic relations of this model are given by the following:

• Definition of potential radius at the top of the boundary layer

  

  

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• Thermal wind balance relation

  

  

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• Cyclostrophic balance relation at the top of the boundary layer

  

  

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• Budget equation for boundary layer entropy of the inner region (R ≤ R_m)

  

  

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• Entropy at the sea surface

  

  

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• Boundary layer entropy of the outer region (R > R_m)

  

  

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  where

  

  

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The indices b, s, t, and a indicate evaluation at the top of the boundary layer, at the sea surface, at the ambient tropopause, and in the boundary layer of the far environment, respectively. The asterisk denotes that the thermodynamic variable is considered at water vapor saturation. All other notation is given in Table 1. In the present study we designate χ = (T_s − T_t)s for simplicity also as entropy. In E86, an atmosphere is assumed that is neutral to slantwise convective instability. Therefore, the saturation entropy χ* is vertically constant above the boundary layer in potential radius space. Equation (4) is based on the slab boundary model introduced by Schubert and Hack (1983), which does not a priori include the assumption of cyclostrophic wind balance.

Table 1.

Notations for the E-PI model.

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The cyclostrophic balance equation has to be integrated radially in order to determine the surface pressure term P_s. However, since r_b is unknown, it has to be eliminated using the thermal wind relation [Eq. (2)]. This leads to

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Integrating this equation outward from R to an ambient radius where P_s as well as χ* vanishes gives

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E86 distinguishes between the inner region (R ≤ R_m) and the outer region (R > R_m). In the inner region the relative humidity decreases radially outward until it attains the ambient value h_a at R = R_m. A constant relative humidity is assumed in the outer region and explained by vertical mixing of boundary layer air with dry air aloft. Hence, the boundary layer entropy of the outer region decreases in the radial direction only because of the increase in surface pressure. A further serious assumption in E-PI theory is that the boundary layer entropy is identical to the saturation entropy of the free atmosphere (i.e., SCAPE vanishes exactly).

For the inner region (R ≤ R_m, indicated by the index i), we have

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and

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These equations can be solved analytically assuming that ; with this assumption, we obtain a single differential equation:

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The solution of this equation reads

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Here, R_o is an integration constant setting the horizontal extent of the cyclone. Note that the horizontal scale of the tropical cyclone is not determined by E-PI theory.

In the outer region (R > R_m, indicated by the index o), the budget equation for boundary layer entropy is not needed since it will be determined by the assumption of a constant relative humidity. Instead, Eq. (6) will be used and therefore the two equations

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and

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form the basis of the solution in the outer region.

Again, the assumption that entropy in the boundary layer and saturation entropy above the boundary layer coincide is applied in E-PI theory (i.e., ). This condition gives the differential equation

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which has the solution

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The maximum tangential wind arises at the radius R = R_m where both solutions are matched. The matching in E-PI theory requires continuity of entropy and tangential wind but not of radial wind (see E86). The first condition leads to

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This equation has two unknowns, χ_om and R_m. The continuity of tangential wind can be established by using the thermal wind relation

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which results by combination of Eqs. (1) and (2). Therefore, we obtain

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Eliminating χ_om by Eq. (18) gives

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Finally, the solutions for entropy and tangential wind read

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where υ_max,E sets E-PI by the formula

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This maximum tangential wind velocity coincides with that obtained by E86 for the case of a vanishing Coriolis parameter [see his Eq. (43)]. Note that neither the maximum wind speed nor the minimum surface pressure depends on the potential radius of maximum wind. This is a peculiar property of vortices in cyclostrophic balance. The scale independence of E-PI is, however, not realistic as the neglected effects of subscale eddy diffusion can have a different impact in different scale ranges. This leads us to a difficult issue since it is not clear what determines the mixing lengths in vortices of different scales. Furthermore, E86 found scale dependence when planetary rotation is not neglected, but it remains weak.

Finally, the tangential wind profile as a function of physical radius r becomes

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where the physical radius of maximum winds r_m is given by

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It can be seen that the profile given by Eq. (25) coincides with that found by E86 [see his Eqs. (47) and (48)] and that the profile is reasonable as long as C_H/C_D < 2 and A_s < 1.

3. Extended PI

In this section we derive an extended-PI model in which hurricane superintensity is induced by (i) nonzero SCAPE outside the eyewall and (ii) consideration of an eye. If SCAPE exists, the condition

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must be fulfilled. It can be assumed that SCAPE is small beneath the steady-state eyewall since air rises with a large vertical velocity in this region. Outside the eyewall, however, SCAPE seems to exist as was found in numerical simulations (Frisius and Hasselbeck 2009) and in observations (e.g., Sheets 1969; Frank 1977; Bogner et al. 2000; Eastin et al. 2005). In the reversible limit case a temperature discontinuity should evolve between the eyewall region and the outer region because of the presence of SCAPE. In reality, thermal energy is radially mixed by subscale eddies so that the discontinuity will be smoothed. However, the wind speed can still be superintense if the gradient of saturation entropy is large enough. The dependence of superintensity on SCAPE can be outlined on the basis of the thermal wind balance Eq. (19), which can be written as follows:

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where E_C ≡ χ_b − χ* can approximately be identified with SCAPE (see Emanuel 1994) supposing that χ_b ≥ χ*. Assuming that the boundary layer entropy profile takes the same form as in the E-PI model, we obtain

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where υ_b,E is the tangential velocity of the E-PI model. Radial integration from the radius of maximum winds outward to an outer radius R_o gives

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where E_Co denotes SCAPE at R = R_o. For the derivation of Eq. (30), it is assumed that SCAPE vanishes at the radius of maximum winds. We see that superintensity can arise at the radius of maximum winds if SCAPE is present outside the eyewall. The smaller the difference between R_m and R_o, the more certain is the occurrence of superintensity in response to ambient SCAPE.

First, we derive superintensity by discarding the dependence of surface entropy on pressure. This can be done by setting A_s to zero. To retain a realistic vortex structure in the outer region, we leave A₀ unchanged since a corresponding radial decay of boundary layer entropy can arise by a radial decrease of relative humidity for A_s = 0. The freedom for solutions is restricted in potential-radius space since a frontal collapse might occur—that is, ∂r_b/∂R < 0. To avoid this, saturation entropy must decrease radially outward by a power of less than 4 (Emanuel 1997). The E-PI model is based on the assumption of a vanishing saturated moist potential vorticity (PV); that is, saturation entropy does not vary along angular momentum surfaces. This may be well suited for the eyewall but not necessarily within the outer region and the eye. In these regions tangential wind and saturation entropy do not need to be related to each other since saturated moist PV can be generated by dry adiabatic warming and mixing of angular momentum. In appendix A an alternative interpretation of the thermal wind balance Eq. (2) is introduced. From this perspective, χ* is defined as a vertical average that still fulfills this equation for the nonuniform PV case. Therefore, we can still use this equation without assuming vanishing PV. We divide the hurricane into three regions, namely the eye (0 < R ≤ R_e), the eyewall (R_e < R ≤ R_m), and the outer region (R > R_m).

Within the eye (index e) we assume a linear increase of tangential wind with radius (solid-body rotation). Therefore, the tangential wind becomes

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where υ_b,e0 is the tangential wind at the rim of the eye (R = R_e). With A_s = 0, the solution of Eq. (4) becomes

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The integration constant R_o is the radius where χ_b,e would become zero. The saturation entropy of the eye is obtained by integration of Eq. (19):

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where is the entropy at the rim of the eye and we require that at R = R_e. Thus we ensure that SCAPE vanishes at the boundary between eye and eyewall. Therefore,

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where δ is a factor with a positive value less than one (0 < δ ≤ 1). It accounts for subsaturation in the boundary layer air and is therefore denoted as the subsaturation factor. With neglected pressure dependence of entropy (A_s = 0), it is related to relative humidity by δ ≃ (h_b,e0 − h_a)/(1 − h_a), where h_b,e0 is the relative humidity in the boundary layer at R = R_e.

In the eyewall (index ew), we assume the following entropy profile:

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where the exponent κ (≤4) determines the radial decrease of entropy. The description of the radial profile by Eq. (35) is predicated on the assumption that the eyewall has the character of a front. The parameter κ determines the sharpness of this front and it may range from a value typical for E-PI theory, where the eyewall has a finite width, up to a value of 4, at which a frontal collapse is implied. For κ = 2C_H/C_D and R_o = R_m0, SCAPE vanishes and the E-PI solution is recovered. We believe that κ is mainly affected by horizontal mixing although we are aware that the radial profile may not reveal the same shape in a tropical cyclone simulated by a complex numerical model. However, the exact shape is not crucial to the conclusion we derive from this study because the limiting case κ = 4 is well defined. It represents an upper intensity bound for the hypothetical situation where no horizontal mixing takes place and the eyewall updraft is infinitely narrow. Furthermore, we note that the profile of the boundary layer entropy does not coincide exactly with the saturation entropy within the eyewall. Therefore, a small amount of SCAPE within the eyewall is inherent to the model. Only at R = R_e are boundary layer entropy and saturated eyewall entropy identical. At larger radii, lower values of eyewall entropy may result from radial mixing. We assume that the entropy profile described by Eq. (35) only extends up to a radius of R_m < R_m0. The tangential wind becomes due to the thermal wind balance Eq. (19)

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where υ_max denotes the maximum tangential wind:

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In the eyewall the boundary layer entropy profile of the eye is continued [see Eq. (32)]:

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To determine R_m0 as a function of R_m, assumptions for the outer region (R > R_m, index o) must be specified. It is a reasonable assumption that saturation entropy describes a profile as in E-PI theory for the outer region (R > R_m). Then, entropy becomes

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where η is the power by which the entropy decays in the outer region. Finally, continuity of entropy and tangential wind leads to the following matching conditions at R = R_m:

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and

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These conditions have the consequence that

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At a certain radius R_a outside the eyewall, becomes equal to .³ Beyond this radius, we assume that entrainment of overlying air leads to a neutral stratification as in E-PI theory, and therefore

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Finally, by substitution of Eq. (42) in Eq. (37) with δ from Eq. (34), we find the following for the maximum tangential wind:

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It is obvious that this formula reduces for A_s = 0 to Eq. (24) as predicted by E-PI theory when R_e = 0 (no eye), R_o ≡ R_m0 (no SCAPE), and η = μ and κ = ν (finite eyewall entropy gradient). Assuming R_e = 0 (no eye) and η = μ, hurricane superintensity arises only for κ > ν. This condition can only be fulfilled if C_H/C_D < 2 where E-PI predicts the formation of a discontinuity. Further superintensity comes into play by the existence of an eye (R_e > 0) in the extended-PI model. The upper bound of intensity results for R_e = R_m, R_o = ∞ and κ = 4. In this extreme case, boundary layer air is in equilibrium with the sea surface and the eyewall collapses to an infinitely thin discontinuity at R = R_m. Air would exit the boundary layer at the radius of maximum winds in a narrow updraft current. Under these conditions, the maximum tangential wind depends only on and η and is given by However, this extreme-case solution is not realistic as it requires the existence of a convectively unstable saturated boundary layer in the far environment [see Eq. (38)].

In the extended-PI model the radial profiles for tangential wind and saturation entropy take the following shape:

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and

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So far, we have excluded the pressure dependence of surface entropy. A derivation of extended PI for surface pressure–dependent entropy is provided in appendix B. The result for maximum tangential wind reads

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It becomes evident that this formula reduces to Eq. (44) for A_s = 0 when Eq. (34) is kept in mind. Using δ = 1 leads in most cases for pressure-dependent surface entropy to unphysical solutions where boundary layer entropy increases with increasing radius in the outer region. Therefore, δ is adjusted numerically so that boundary layer entropy and saturation entropy of the overlying atmosphere coincide at R_a ≥ R_m. The factor δ can become less than unity for two reasons: 1) inflowing air does not gather enough moisture from the sea surface to reach saturation and/or 2) the sensible heat flux from the ocean surface is not large enough to compensate the temperature decrease by adiabatic expansion of the inward flowing air. Indeed, the numerical simulations presented in section 4b suggest that the second reason is more important. However, the fundamental Eqs. (3)–(5) are based on the idealized assumption of an isothermic boundary layer. In both E-PI and the extended-PI model a small error is introduced by this assumption.

4. Results

a. Results from the extended-PI model

This section presents results from the extended-PI model. As standard model parameters we assume C_H/C_D = 1, T_s = 301.15 K, T_t = 221.15 K, h_a = 0.7, ζ₀ = 0.5 × 10⁻⁵ s⁻¹, and R_d = 287.07 J kg⁻¹ K⁻¹. The model has essentially four unknown parameters R_e/R_m, R_a/R_m, κ, and η, which set the eye radius, the amount of SCAPE, radial entropy gradient within the eyewall, and the radial decay of entropy outside the eyewall, respectively. At present we do not have a theory for these parameters. However, we can assume that horizontal eddy diffusion smoothes the radial profile and therefore weakens hurricane intensity. This effect may result in a smaller κ. We assume η = μ for the solutions since it leads to a suitable vortex outside the eyewall. Note that there is no limit in the choice of η, but it becomes questionable to assume very large η in the context of a balanced boundary layer model because then the radial wind speed becomes unrealistically intense.

Figure 2 displays profiles of saturation entropy, tangential wind, and SCAPE as a function of potential radius R for different κ values and a pressure-independent surface entropy (A_s = 0). For the plotted solutions we assumed that R_e = 150 km and R_m0 = (ν/κ)^1/2R_a with R_a = 300 km. It is clear from Fig. 2 that superintensity increases with increasing SCAPE. As κ decreases, the solution approaches the E-PI solution (κ = 2) where the boundary layer entropy and saturation entropy are identical and therefore SCAPE vanishes. For R ≤ R_a, the boundary layer entropy remains unchanged with increasing κ. However, the overlying atmosphere exhibits increasingly smaller saturation entropy values and hence an increase of SCAPE, which can be seen from Fig. 2c. As a consequence, we obtain a larger entropy gradient and therefore enhanced tangential wind within the eyewall. Because of a nonvanishing eye radius, saturation entropy in the eye significantly exceeds the surface value. In turn, the eye is appreciably warmer than what would occur from moist adiabatic ascent. This has been noted for some time by Riehl (1954) for real tropical cyclones and explained by Smith (1980) and Emanuel (1997) on the basis of thermal wind balance. They showed that a radially inward increase of saturation entropy must appear since the eye is in solid-body rotation. For κ = 2, the increase is exactly identical to that of the boundary layer entropy beneath, whereas for larger κ values the increase becomes more intense above the boundary layer. Figure 3 displays the same profiles but as a function of physical radius r instead of R. In these plots the eyewall region can be identified more clearly and the decrease of its width with increasing κ becomes evident. SCAPE declines rapidly from the center toward the rim of the eye. Furthermore, it can be seen that the radial shear of the tangential wind in the eyewall rapidly grows as κ increases and becomes infinitely large for κ = 4 when the eyewall takes the form of a discontinuity.

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Profiles of the extended-PI model with pressure-independent surface entropy for different κ values (see legend): (a) saturation entropy, (b) tangential wind, and (c) SCAPE as a function of potential radius. Solutions have been calculated for R_e = 150 km, R_o = 300 km, and R_m0 = (ν/κ)^1/2R_a. Note that the profiles for κ = 2 are associated with the E-PI solution.

Citation: Journal of the Atmospheric Sciences 69, 2; 10.1175/JAS-D-11-064.1

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As in Fig. 2, but profiles are shown as a function of physical radius.

Citation: Journal of the Atmospheric Sciences 69, 2; 10.1175/JAS-D-11-064.1

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The effect of the pressure-dependent surface entropy is illustrated in Figs. 4 and 5. Figure 4 displays boundary layer entropy and saturation entropy of the free atmosphere as a function of physical radius and the corresponding entropy profile of the E-PI solution. When the pressure effect is not neglected, entropy in the core of the TC is considerably larger. At the vortex axis, the value is almost twice as high. Furthermore, the vortex shrinks in its radial extent because of the enhanced tangential wind (Fig. 5). Not only the maximum tangential wind but also the superintensity (Δυ = υ_max − υ_max,E) becomes larger when the pressure dependence of surface entropy is taken into account. Furthermore, the shape of the tangential wind profile in the inner region is different in the E-PI solution for a pressure-dependent surface entropy but not for the extended-PI solution since the tangential wind variation with radius is a prescribed part of the solution.

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Boundary layer entropy (solid line) and saturation entropy (dotted line) of the extended-PI model in comparison to the E-PI solution (dashed line) for κ = 3, R_e = 150 km, R_a = 300 km, and R_m0 = (ν/κ)^1/2R_a as a function of physical radius, showing solutions for (a) pressure-independent surface entropy (A_s = 0) and (b) pressure-dependent surface entropy (A_s = 0.45).

Citation: Journal of the Atmospheric Sciences 69, 2; 10.1175/JAS-D-11-064.1

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Tangential wind of the extended-PI solution (solid line) and the E-PI solution (dashed line) for κ = 3, R_e = 150 km, R_a = 300 km, and R_m0 = (ν/κ)^1/2R_a as a function of physical radius, showing solutions for (a) pressure-independent surface entropy (A_s = 0) and (b) pressure-dependent surface entropy (A_s = 0.45).

Citation: Journal of the Atmospheric Sciences 69, 2; 10.1175/JAS-D-11-064.1

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In the cases discussed above, χ_b ≥ χ* is fulfilled in the eyewall region. However, for smaller eye diameters (e.g., R_e = 25 km) this condition is not fulfilled anymore since the χ_b and χ* profiles intersect at three radii (Fig. 6a) instead of two as in Fig. 4. This result is unphysical because it requires an artificial heat source in the eyewall. To gain a more realistic result, we assume that the limiting case—for which χ_b(R_ew) = χ*(R_ew) and ∂χ_b(R_ew)/∂R = ∂χ*(R_ew)/∂R—holds at a certain radius R_ew within the eyewall region. Then, the associated curves would be tangent to each other at R = R_ew (Fig. 6b). Consequently, the intensity at R_ew would be identical to the solution for which R = R_ew is assumed. The uncorrected solution with the boundary condition χ_b(R_e) = χ*(R_e) is shown in Fig. 6a. It is clear that χ_b is smaller than χ* throughout a large part of the eyewall region leading to nonzero SCAPE in the eye boundary layer since χ_b > χ* for R < R_e. For the corrected solution that fulfills the above-described conditions (Fig. 6b), we find that the saturation entropy of the free atmosphere is lower in the core than in the uncorrected version. This results in a reduced vortex intensity (the maximum tangential wind reaches 86.7 instead of 92.5 m s⁻¹).

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Boundary layer entropy (solid line), saturation entropy (dotted line), and tangential wind (dashed line) as a function of potential radius for R_e = 25 km and other parameters as for the solution displayed in Figs. 4 and 5. (a) Uncorrected solution. (b) The solution has been corrected in such a way that the inequality χ_b ≥ χ* is just fulfilled in the eyewall region.

Citation: Journal of the Atmospheric Sciences 69, 2; 10.1175/JAS-D-11-064.1

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Since SCAPE > 0 within the eye for R_e = 25 km, the solutions for larger eye diameters seem to be more realistic (cf. Figs. 4a and 6). Hurricane superintensity given by the expression Δυ = υ_max − υ_max,E and the subsaturation factor δ as a function of R_e/R_m and κ are shown in Fig. 7 for the uncorrected solution (Figs. 7a,c) and the corrected solution (Figs. 7b,d) with no more than two intersection points along the radial profile (see above). For relatively large eye diameters, with only two intersection points, both solutions are identical. For smaller eye diameters, the correction results in an independence of superintensity with respect to eye diameter. It can be seen that hurricane superintensity generally increases with increasing κ (Figs. 7a,b). The largest superintensity of more than 35 m s⁻¹ arises for the largest eye diameter and κ = 4. For a decreasing eye radius along a fixed κ value, intensity weakens until it attains a minimum, and then increases again (Fig. 7a). The behavior looks different for the corrected solution. At smaller eye radii beyond the minimum, superintensity remains constant and does not rise again (Fig. 7b). The subsaturation factor δ decreases with increasing eye diameter and decreasing κ in both versions (Figs. 7c,d).

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(a),(b) Hurricane superintensity υ_max − υ_max,E (m s⁻¹) and (c),(d) subsaturation factor δ as a function of κ and R_e/R_m for R_a = 300 km and R_m0 = (ν/κ)^1/2R_a. Results shown in (b) and (d) correspond to those in (a) and (c), but the solution has been corrected in such a way that the inequality χ_b ≥ χ* is just fulfilled in the eyewall region.

Citation: Journal of the Atmospheric Sciences 69, 2; 10.1175/JAS-D-11-064.1

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When R_a approaches R_m, SCAPE drops and becomes zero for R_a = R_m. Hence, superintensity should vanish in this limit case. As demonstrated in Fig. 8, showing superintensity as a function of R_a/R_m and R_e/R_m, the solution must be corrected in the aforementioned way to ensure this. Obviously, superintensity only approaches zero in the corrected solution. Furthermore, it can be seen that superintensity approaches a constant value for a fixed eye radius when R_a is increased. We can conclude that hurricane superintensity depends on the existence of SCAPE in the present model when we require that χ_b ≥ χ* in the eyewall. Otherwise, an unrealistic heat source arises in the eyewall, leading to superintensity even for vanishing SCAPE.

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(a) Hurricane superintensity υ_max − υ_max,E (m s⁻¹) as a function of R_a/R_m and R_e/R_m for κ = 4. (b) As in (a), but the solution has been corrected in such a way that the inequality χ_b ≥ χ* is just fulfilled in the eyewall region.

Citation: Journal of the Atmospheric Sciences 69, 2; 10.1175/JAS-D-11-064.1

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b. Results from the numerical HURMOD simulations

To substantiate our results we have evaluated numerical experiments with the cloud-resolving axisymmetric model HURMOD. This model has been described by Frisius and Hasselbeck (2009) and Frisius and Wacker (2007). Here, we repeat an experiment using the same initial state and boundary conditions as in the sensitivity experiment NOEVAP [for details, see Frisius and Hasselbeck (2009)]. In this experiment the simulated tropical cyclone attains a quasi-steady state and therefore is suitable for comparison with the present analytical PI model of a steady-state tropical cyclone. Yet, some modifications to model physics and resolution were necessary to simulate a tropical cyclone with significant superintensity. The first modification is that condensed water is immediately removed, which leads to a larger growth rate and a higher maximum intensity. BR09 also made use of this simple cloud scheme as their PI model was related to pseudoadiabatic ascent in the eyewall. Other differences regarding model resolution and model physics are described in appendix C. The initial state is prescribed by a horizontally uniform atmosphere with a superimposed initial vortex with 6 m s⁻¹ maximum wind speed at the surface, a relative humidity of 70%, and a tropospheric temperature lapse rate of 0.0065 K m⁻¹ up to 10-km height. With this stratification, the initial ambient atmosphere contains CAPE and its initial value amounts to 1098 J kg⁻¹ when condensate loading is neglected. This assumption is consistent with the simplified cloud scheme we apply here. We are aware that E-PI and extended PI is connected with zero environmental CAPE. However, no matter whether there is CAPE in the initial state or not, the far environment exhibits convective stability after the tropical cyclone equilibrates except near the outer model boundary where a narrow sponge is introduced as a standard method to damp high-frequency waves.

We performed three experiments. As a standard or reference case, we take the experiment described in the preceding paragraph and refer to it as REF. In the second experiment, denoted as NEUTRAL, the initial state is neutrally stratified. This is accomplished by setting the initial temperature profile to a pseudoadiabat. Furthermore, the simplified radiation scheme of HURMOD relaxes the temperature profile to a neutral state instead of an unstable one as in REF. Therefore, steady-state solutions of NEUTRAL and REF can have quite different intensities, if they exist. The third experiment, denoted as HORDIFF, is identical to REF except for enhanced horizontal diffusion. Additional horizontal diffusion as described by Rotunno and Emanuel (1987) is implemented in simulation HORDIFF where the mixing length of the additional diffusion is l_h = 600 m² s⁻¹.

The time development of maximum tangential wind computed from the HURMOD experiments is displayed in Fig. 9. A rapid and continuous growth stage begins after a short period of several convective bursts in REF and HORDIFF while NEUTRAL exhibits a slower intensification rate. The phase of rapid intensification ends after 18 h in REF and HORDIFF and after 50 h in NEUTRAL. Afterward, the cyclone intensity varies slowly and all solutions become quasi-steady. The most intense tropical cyclone is simulated by REF with a maximum wind speed of slightly more than 100 m s⁻¹. Experiments NEUTRAL and HORDIFF have a smaller maximum wind speed of approximately 85 and 82 m s⁻¹, respectively.

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Maximum horizontal wind speed from the HURMOD simulations REF (solid line), NEUTRAL (dashed line), and HORDIFF (dotted line) as a function of time.

Citation: Journal of the Atmospheric Sciences 69, 2; 10.1175/JAS-D-11-064.1

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Looking at the radial profiles of tangential wind and gradient wind in the mature state at about 1-km height (Fig. 10), we see that a significant gradient wind imbalance occurs at the radius of maximum tangential wind, whereas the imbalance is negligible at the radius of maximum gradient wind. This is in agreement with Fig. 3 of BR09. Overall, these profiles are qualitatively similar to those of BR09 so that their simulation results do not appear to be model specific. The figure also displays the tangential wind of E-PI theory. Instead of the full solution given by Eq. (24) we calculated E-PI by the formula

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which can be obtained by combining Eqs. (1), (2), and (4). This equation is more appropriate for analyzing model output since it is difficult to assess a suitable boundary layer relative humidity h_a. We used T_s = 301.15 K and T_t = 238 K. The latter has been estimated from HURMOD simulations as the characteristic outflow temperature. E-PI varies between 50 and 63 m s⁻¹ in the eyewall region. Outside this region the formula is not applicable. Hence, the simulated tropical cyclones are clearly superintense. The HURMOD simulations we performed result in different values for superintensity with respect to the calculated E-PI. In REF, about 22 m s⁻¹ can be explained on the basis of gradient wind and another 14 m s⁻¹ arises by gradient wind imbalance. In NEUTRAL and HORDIFF superintensity is smaller. The widest eyewall and largest eye diameter arise for HORDIFF, while NEUTRAL has the smallest eye and eyewall width.

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Radial profiles at z = 1062.5 m of tangential wind (filled circles) and gradient wind (open squares) averaged over the time period of 130–140 h for different HURMOD experiments: (a) REF, (b) NEUTRAL, and (c) HORDIFF. Estimated E-PI (open circles) is displayed for comparison. The vertical lines enclose the eyewall region where vertical velocity at z = 1062.5 m is larger than 0.5 m s⁻¹.

Citation: Journal of the Atmospheric Sciences 69, 2; 10.1175/JAS-D-11-064.1

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To support our analytical model we have to verify that SCAPE occurs in HURMOD simulations. For this purpose SCAPE has been analyzed in the three simulations. As start conditions for the SCAPE calculation we choose the average values between the first and second model levels. SCAPE values are only considered when a level of no buoyancy appears for heights larger than 10 km (deep convection). Figure 11 shows SCAPE as a function of potential radius for the three simulations at t = 140 h. The vertical lines mark the position of the eyewall. Consistent with the extended-PI model maximum SCAPE appears outside the eyewall and becomes rapidly zero in the eyewall region. In REF and HORDIFF, the amount of SCAPE is similar and it decays radially vanishing at R ≈ 550 km. In NEUTRAL, a much smaller amount of SCAPE is present and it goes to zero at R ≈ 310 km. This is consistent with the smaller superintensity compared to experiment REF. On the other hand, the similarity of SCAPE profiles of REF and HORDIFF in this figure seems to disagree with the different superintensities of gradient wind in these experiments. However, maximal superintensity arises in agreement with Eq. (29) at the outer eyewall boundary where the radial SCAPE gradient becomes large. Here, the difference in REF and HORDIFF is clearly visible when we look at the profiles in detail (Fig. 11d). The SCAPE gradient is smaller in HORDIFF than in REF. Direct numerical evaluation of Eq. (29) gives larger values of superintensity (75, 60, and 55 m s⁻¹ for REF, NEUTRAL, and HORDIFF, respectively). This overestimation of gradient wind by Eq. (29) hints at a problem with the underlying assumptions of this equation. The quantitative difference between the predicted [Eq. (29)] and simulated superintensity values may also result from the occurrence of regions with negative absolute vorticity just outside the eyewall where a discontinuity forms in potential radius space. Such inertially unstable regions have also been found by BR09 and may lead to an overestimation of SCAPE since the vertical integration in potential radius space is across the discontinuity. This problem cannot be resolved here and needs further investigation. However, we find a qualitative agreement in the relationship between the radial SCAPE gradient and superintensity of gradient wind.

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SCAPE (solid lines) at 140 h as a function of potential radius for different HURMOD experiments: (a) REF, (b) NEUTRAL, and (c) HORDIFF. The vertical lines enclose the eyewall region where vertical velocity at z = 4937.5 m is larger than 0.5 m s⁻¹. (d) An enlarged view covering the eyewall region for REF (solid line), NEUTRAL (dashed line), and HORDIFF (dotted line).

Citation: Journal of the Atmospheric Sciences 69, 2; 10.1175/JAS-D-11-064.1

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Radial profiles of vertically averaged saturation entropy obtained by vertical integration from z = 1000 to 11 000 m,⁴ entropy at the lowest level (z = 62.5 m), and entropy at the sea surface are displayed in Fig. 12. Comparing results of the extended-PI model (see Figs. 3–6), we find similarities but also differences. Outside the eyewall (indicated by vertical lines), boundary layer entropy is larger than vertically averaged saturation entropy in the free atmosphere due to nonzero SCAPE (cf. Fig. 11). In all HURMOD simulations, the boundary layer entropy is much smaller than the sea surface entropy in the eyewall region. The main reason for this is the temperature decrease of up to 4 K below sea surface temperature. We see that the assumption of isothermal inflow made by E86 is clearly not fulfilled in the simulated superintense tropical cyclone. In the eye, boundary layer entropy increases further so that it attains its maximum here. The entropy difference between the inner and outer edge of the eyewall is smaller in NEUTRAL and HORDIFF. This is consistent with the intensity difference between the two experiments. In all simulations, saturation entropy is much larger in the eye than in the eyewall. In agreement with previous findings (Riehl 1954; Emanuel 1997), it even exceeds the surface value.

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Profiles of vertically averaged saturation entropy of the free atmosphere (solid line), boundary layer entropy s_b at z = 62.5 m (dotted line), and sea surface entropy s_s (dashed line) as a function of potential radius for the HURMOD simulations (a) REF, (b) NEUTRAL, and (c) HORDIFF at t = 140 h. The vertical lines enclose the eyewall region where vertical velocity at z = 4937.5 m is larger than 0.5 m s⁻¹.

Citation: Journal of the Atmospheric Sciences 69, 2; 10.1175/JAS-D-11-064.1

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Some parameters deduced from Fig. 12 are listed in Table 2 for the different HURMOD experiments. In general, the ratio R_e/R_m takes large values about 0.8. The slope of the profile in the eyewall does not show the exact shape as given by Eq. (46) and therefore we cannot provide a useful estimate for κ. However, it can be seen that the negative slope tends to steepen with increasing potential radius in the eyewall region, which is consistent with the extended-PI model. This is also consistent with the result that the maximum gradient wind appears at or near the outer edge of the eyewall (cf. Fig. 10). The ratio R_a/R_m of 1.625 is notably smaller in NEUTRAL than in the other experiments because of the small amount of SCAPE. Table 2 also contains the subsaturation factor δ and superintensity of gradient wind determined from the extended-PI model by assuming κ = 2.5 and using the estimated parameters R_e, R_m, and R_a. It becomes evident that the analytical model predicts larger values for δ in each case. But the differences between the different experiments are in qualitative agreement with the lowest value appearing for δ in experiment NEUTRAL. The superintensities of gradient wind in REF and NEUTRAL are roughly similar in both models, but highly different for HORDIFF. HURMOD returns the lowest superintensity of gradient wind, whereas the extended-PI model returns the largest superintensity for HORDIFF. This is probably due to the prescription of κ by a fixed value. Because horizontal diffusion is implemented, a smaller value should be chosen for the comparison with HORDIFF. This becomes evident from Figs. 12a,c. It shows that the radial gradient of in the eyewall is lower in HORDIFF than in REF.

Table 2.

Estimation of some quantities from the HURMOD experiments. The parameter υ_max,g − υ_max,E denotes the superintensity of the gradient wind in HURMOD at z = 1062.5 m; δ_analyt and υ_max,analyt are the subsaturation factor and maximum tangential wind, respectively, estimated from the extended-PI model by using the respective values for R_e, R_m, and R_a and assuming κ = 2.5.

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5. Discussion

It is worthwhile to compare the present approach to superintensity with previous theories. In essence, superintensity has been explained in the scientific community by two different theories. The first theory, by Persing and Montgomery (2003), suggests that superintensity results from the injection of high-entropy air from the eye into the eyewall. Bryan and Rotunno (2009b) demonstrated by sensitivity experiments with their cloud-resolving axisymmetric model that this flux has a negligible impact on superintensity. They switched off the surface enthalpy flux in the eye, which leads to smaller entropy in the lower eye but not to a weaker intensity. However, the eye may still have an effect since it is possibly warmer than the eyewall because of the occurrence of subsidence and radial turbulent transfer. In the extended-PI model, eyewall warming by heat transfer from the eye is suppressed by setting boundary layer entropy larger or equal to eyewall saturation entropy of the free atmosphere in the eyewall region. HURMOD simulations do not suggest significant heating by the eye since is similar to the boundary layer entropy in the inner part of the eyewall.

The second mechanism for superintensity was studied by BR09 who introduced an analytical model in which gradient wind imbalance explains superintensity. The superintensity results from the inertia of the inflowing boundary layer air that overshoots the radius of exact gradient wind balance. They compared the results with the output of an axisymmetric hurricane model and found a surprisingly good agreement. Their analytical formula for maximum wind speed [their Eq. (23)] contains an additional term that describes the impact of gradient wind imbalance. They relate E-PI (i.e., PI of gradient wind) to a different formula than in E86. It is identical to Eq. (48) except for an additional factor T_s/T_t multiplied on the right-hand side of Eq. (48) in BR09, which stems from dissipative heating in the boundary layer (see Bister and Emanuel 1998). This process is neglected here. BR09 determined χ_b from model output at the radius of maximum wind. Therefore, they do not present a closed PI model that should only depend on environmental parameters as in E86. In contrast, we relate E-PI to Eq. (24) in which only environmental parameters appear. Our extended model can explain superintensity of gradient wind in terms of Eq. (48) only by the existence of SCAPE at the radius of maximum gradient wind (i.e., χ_b > χ*). This is confirmed by the results of HURMOD experiments (see Figs. 11 and 12). To demonstrate that the extended-PI model also exhibits superintensity in terms of Eq. (48) we calculated υ_max,E as given by this equation using the boundary layer entropy and surface pressure from the solution shown in Fig. 5b. Figure 13 displays the result together with the full E-PI tangential wind profile. We see that E-PI in terms of Eq. (48) is even smaller than in the complete E-PI model. Note that Eq. (48) is not valid in the outer region (R > R_m) of the E-PI model because the boundary layer entropy budget Eq. (4) is replaced by Eq. (6) in this region. For this reason, the profile exhibits a very unrealistic shape in the outer region.

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The tangential wind profile determined from Eq. (48) (solid line) in comparison with the tangential wind profile of the full E-PI solution (dashed line). The values used in Eq. (48) were taken from the case shown in Figs. 4b and 5b.

Citation: Journal of the Atmospheric Sciences 69, 2; 10.1175/JAS-D-11-064.1

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The role of CAPE has also been discussed in previous PI studies. Holland (1997) provided a PI theory in which environmental CAPE can be included. In his model, it is assumed that the cloud base of the eyewall lies near the sea surface and that entropy is vertically constant in the eyewall as well as in the subsaturated eye. His model can predict surface pressure but it does not provide a prediction of the radial temperature gradient that actually governs the intensity in terms of wind speed in pressure coordinates [cf. Eq. (19)]. Both Camp and Montgomery (2001) and Persing and Montgomery (2005) doubt the relevance of environmental CAPE for PI. In the latter study it was demonstrated by an axisymmetric model that superintensity is quite independent of environmental CAPE. So, one may question the relevance of the extended-PI model since CAPE is a crucial element of the model. However, the nonexistence of CAPE in the far environment does not exclude the possibility of existing CAPE or SCAPE in the region immediately outside the eyewall. Therefore, we see no contradiction to the findings of these studies.

Finally, one could argue that our superintensity model is not valid for energetic reasons. E86 and Emanuel (1997) demonstrated that the E-PI model can be understood as a Carnot engine. Therefore, E-PI would be associated with the maximum efficiency of a heat engine and cannot be larger because of the second law of thermodynamics. However, Wang and Xu (2010) stated that E-PI given by Eq. (48) only results after questionable approximations from the Carnot cycle model. Emanuel (1997) assumed that most of the surface heat flux arises near the radius of maximum winds. Nevertheless, Wang and Xu (2010) found in a tropical cyclone simulation that the radial entropy flux beneath the eyewall is much larger than the surface vertical entropy flux. Reducing the horizontal flux by switching off evaporation outside the eyewall leads to a significantly reduced intensity. This hints at the existence of CAPE outside the eyewall in agreement with our ideas. Furthermore, the view of a Carnot cycle may not be valid since air leaving the tropical cyclone in the outflow may not enter it again. Possibly, it is more appropriate to view a tropical cyclone as a flow heater in which inflowing and outflowing air fulfill different boundary conditions at large radii. This is indeed the case for the E-PI model although the predicted intensity is consistent with the Carnot cycle model.

6. Conclusions

In this study we developed and analyzed an extended model for potential intensity of tropical cyclones. It is based on Emanuel’s PI model but it relaxes the condition of vanishing SCAPE outside the eyewall and it additionally includes an eye. Both features may explain the hurricane superintensity observed in high-resolution numerical models. Additional SCAPE outside the eyewall enhances the radial eyewall temperature gradient and, thereby, the tangential wind. The existence of an eye can also contribute to higher intensity. The superintensity can reach up to 50 m s⁻¹ when SCAPE and the eye diameter become large. SCAPE outside the eyewall is a prerequisite for superintensity when we require that boundary layer entropy is always greater than that of the overlying free atmosphere within the eyewall. Otherwise, a heat source would have to appear in the eyewall, which is questionable.

Numerical simulations of superintensity by HURMOD give further evidence for the importance of SCAPE, beside supergradient winds as discovered previously by BR09. It is found that the boundary layer entropy is not able to attain values comparable to saturation entropy at the sea surface. Furthermore, the boundary layer air does not expand isothermally. Therefore, the Carnot cycle concept of E86 seems to be badly fulfilled in HURMOD simulations. Instead, generation of SCAPE delivers an explanation for the superintensity as observed in our model experiments. This conclusion is also supported by a study of Wang and Xu (2010). Further sensitivity studies of superintensity in high-resolution models are necessary in the future to manifest the respective roles of eye dynamics and SCAPE.

In the extended-PI model, we ignored gradient wind imbalance. However, we note that the budget equation for boundary layer entropy [Eq. (4)] is also valid when the gradient wind balance is violated. Indeed the solutions of the E-PI and extended-PI model for the boundary layer entropy would also result in an unbalanced model if we ignore that the surface pressure occurring in the expression for saturation entropy at the surface is based on the gradient wind balance approximation. Then, an additional superintensity can be calculated by the determination of the unbalanced boundary layer wind. This could be done with a slab boundary layer model like the one employed by Smith et al. (2008) to study the boundary layer of a prescribed E-PI vortex. The investigation of this effect will be subject of a future study.