• Bell, M. M., and M. T. Montgomery, 2008: Observed structure, evolution, and potential intensity of category 5 Hurricane Isabel (2003) from 12 to 14 September. Mon. Wea. Rev., 65, 20252046, https://doi.org/10.1175/2007MWR1858.1.

    • Search Google Scholar
    • Export Citation
  • Bryan, G. H., and R. Rotunno, 2009: Evaluation of an analytical model for the maximum intensity of tropical cyclones. J. Atmos. Sci., 66, 30423060, https://doi.org/10.1175/2009JAS3038.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Emanuel, K. A., 1986: An air–sea interaction theory for tropical cyclones. Part I: Steady-state maintenance. J. Atmos. Sci., 43, 585604, https://doi.org/10.1175/1520-0469(1986)043<0585:AASITF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Emanuel, K. A., 1995: Sensitivity of tropical cyclones to surface exchange coefficients and a revised steady-state model incorporating eye dynamics. J. Atmos. Sci., 52, 39693976, https://doi.org/10.1175/1520-0469(1995)052<3969:SOTCTS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Emanuel, K. A., 2012: Self-stratification of tropical cyclone outflow. Part II: Implications for storm intensification. J. Atmos. Sci., 69, 988996, https://doi.org/10.1175/JAS-D-11-0177.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Emanuel, K. A., and R. Rotunno, 2011: Self-stratification of tropical cyclone outflow. Part I: Implications for storm structure. J. Atmos. Sci., 68, 22362249, https://doi.org/10.1175/JAS-D-10-05024.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hakim, G. J., 2011: The mean state of axisymmetric hurricanes in statistical equilibrium. J. Atmos. Sci., 68, 13641376, https://doi.org/10.1175/2010JAS3644.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kepert, J. D., 2006: Observed boundary-layer wind structure and balance in the hurricane core. Part II. Hurricane Mitch. J. Atmos. Sci., 63, 21942211, https://doi.org/10.1175/JAS3746.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Montgomery, M. T., and R. K. Smith, 2017: Recent developments in the fluid dynamics of tropical cyclones. Annu. Rev. Fluid Mech., 49, 541574, https://doi.org/10.1146/annurev-fluid-010816-060022.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 13351347, https://doi.org/10.1175/BAMS-87-10-1335.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Montgomery, M. T., J. A. Zhang, and R. K. Smith, 2014: An analysis of the observed low-level structure of rapidly intensifying and mature Hurricane Earl (2010). Quart. J. Roy. Meteor. Soc., 140, 21322146, https://doi.org/10.1002/qj.2283.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Persing, J., M. T. Montgomery, J. McWilliams, and R. K. Smith, 2013: Asymmetric and axisymmetric dynamics of tropical cyclones. Atmos. Chem. Phys., 13, 12 29912 341, https://doi.org/10.5194/acp-13-12299-2013.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Rousseau-Rizzi, R., and K. Emanuel, 2019: An evaluation of hurricane superintensity in axisymmetric numerical models. J. Atmos. Sci., 76, 16971708, https://doi.org/10.1175/JAS-D-18-0238.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sanger, N. T., M. T. Montgomery, R. K. Smith, and M. M. Bell, 2014: An observational study of tropical cyclone spinup in Supertyphoon Jangmi from 24 to 27 September. Mon. Wea. Rev., 142, 328, https://doi.org/10.1175/MWR-D-12-00306.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Schecter, D. A., 2013: Relationships between convective asymmetry, imbalance and intensity in numerically simulated tropical cyclones. Tellus, 65A, 20168, https://doi.org/10.3402/tellusa.v65i0.20168.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Smith, R. K., and M. T. Montgomery, 2008: Balanced boundary layers used in hurricane models. Quart. J. Roy. Meteor. Soc., 134, 13851395, https://doi.org/10.1002/qj.296.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Smith, R. K., M. T. Montgomery, and S. Vogl, 2008: A critique of Emanuel’s hurricane model and potential intensity theory. Quart. J. Roy. Meteor. Soc., 134, 551561, https://doi.org/10.1002/qj.241.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zhang, J. A., R. F. Rogers, D. S. Nolan, and F. D. Marks, 2011: On the characteristic height scales of the hurricane boundary layer. Mon. Wea. Rev., 139, 25232535, https://doi.org/10.1175/MWR-D-10-05017.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • View in gallery

    Revised cartoon of the proposed differential Carnot cycle that respects the continuity equation. The yellow region depicts the primary eyewall cloud and upper-level cirrus cloud. See text for details.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 152 152 26
PDF Downloads 104 104 13

Comments on “An Evaluation of Hurricane Superintensity in Axisymmetric Numerical Models”

View More View Less
  • 1 Department of Meteorology, Naval Postgraduate School, Monterey, California
  • | 2 Meteorological Institute, Ludwig-Maximillan University, Munich, Germany
© Get Permissions
Open access

Denotes content that is immediately available upon publication as open access.

© 2020 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: Michael T. Montgomery, mtmontgo@nps.edu

The original article that was the subject of this comment/reply can be found at http://journals.ametsoc.org/doi/abs/10.1175/JAS-D-18-0238.1.

Denotes content that is immediately available upon publication as open access.

© 2020 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: Michael T. Montgomery, mtmontgo@nps.edu

The original article that was the subject of this comment/reply can be found at http://journals.ametsoc.org/doi/abs/10.1175/JAS-D-18-0238.1.

1. Introduction

In a recent paper, Rousseau-Rizzi and Emanuel (2019, henceforth RE19), develop a new theory for the potential intensity (PI) of an axisymmetric tropical cyclone, which predicts the maximum tangential wind component at the surface (nominally 10 m above mean sea level). The new formulation, which is referred to as PIs, is a descendant of the pioneering formulation of Emanuel (1986, henceforth E86). The theory appears to be constructed using largely reversible thermodynamics and is contrasted with a corresponding formulation for the potential intensity of the gradient wind (PIg). These two formulations are presented alongside a summary of the pseudoadiabatic formulation of the potential intensity of the (total) azimuthal wind as developed by Bryan and Rotunno (2009, henceforth BR09).

RE19 provide an assessment of the three formulations on the basis of moderately high resolution (2-km radial grid spacing), axisymmetric, numerical simulations using two different numerical models. While the agreement between the new theory (PIs) and the numerical simulations looks persuasive in a first pass, after some reflection there are elements of the derivations that we find puzzling. In particular, the cartoon on which the theory is based does not respect the continuity equation and as discussed later, an attempt to rectify this issue would seem to invalidate the possibility of constructing a differential Carnot cycle on which the new formulation of surface PI is based. The purpose of this comment is to articulate our questions/concerns in the hope of obtaining a better understanding of the basis for the new PI formulations.

2. Observational evidence of superintense storms

Before presenting our questions on the potential intensity formulations, we would like to draw attention to their introduction, where RE19 review findings demonstrating that horizontal diffusion and three-dimensional effects can significantly reduce the maximum intensity relative to the nearly inviscid, maximum intensity of axisymmetric hurricane simulations.1 RE19 (p. 1698) note that “some storms are observed to have winds that are supergradient by up to 10 m s−1.” The quoted statement would seem to imply that the error between gradient wind potential intensity and observations is not known to exceed 10 m s−1. However, we would like to respectfully point out for the record that Hurricane Isabel (2003) was a noteworthy example of an intense storm whose tangential wind exceeded its a priori gradient wind potential intensity for three consecutive days as documented by Bell and Montgomery (2008). A summary of the potential intensity estimates and estimation caveats is given by Montgomery and Smith (2017), who noted that on 13 September, “the discrepancy, approximately 20 m s−1, between theory and observations spans at least two intensity categories on the Saffir-Simpson Hurricane Wind Scale (Categories 3–5).”

3. Gradient wind PI

In the new derivation of the maximum gradient wind (PIg), in their Eq. (7), RE19 make a few assumptions, both explicit and tacit, that we are particularly puzzled with:

  • Is reversible or irreversible thermodynamics being assumed in this PI derivation? We were not clear about this.
  • The formulation of the friction layer ignores the nonlinear acceleration in the radial direction, which is needed to accurately determine where inflowing moist air parcels rapidly decelerate, turn upward, and ascend out of the boundary layer. As a result, the gradient wind PI formulation presented herein does not determine the radius of maximum tangential wind, an important quantity to know for a tropical cyclone forecaster. The omission of the full radial momentum equation and the silence about the radii of maximum wind and updraft are limitations of the gradient wind formulation that concern us.
  • The friction-layer formulation appears to tacitly assume that the boundary layer flow is well mixed in both absolute angular momentum (hence tangential velocity) and moist entropy sb at leading order since V10 is set equal to Vg,b, and |V10|2 is set equal to Vg,b2, where sb denotes the boundary layer moist entropy per unit mass, Vg,b denotes the gradient wind at the top of the boundary layer and |V10| denotes the surface momentum per unit mass (at 10-m height).2 Since, according to boundary layer theory in the limit of large Reynolds number, the radial pressure gradient is approximately uniform in the boundary layer, the implication is that the boundary layer is tacitly in gradient wind balance also. Since the boundary layer flow owes its existence to the imbalance in the sum of centrifugal, Coriolis, and pressure gradient forces per unit mass, the near gradient-wind balance property of this formulation would seem to be a worry.

4. Azimuthal wind PI

In their summary of BR09’s analytical model for the azimuthal wind (PIa) given by their Eq. (8), RE19 describe the model as a formulation “that accounts for the supergradient contribution” and say also that this formulation represents “a bound for the maximum azimuthal wind.” We have a few questions and comments about the accuracy of these characterizations of the BR09 theory:

  • The boundary layer height is defined to be the height of the maximum tangential wind (BR09, p. 3045). However, high-resolution dropwindsonde observations of intense hurricanes indicate that the maximum tangential wind for an intense storm is generally located at heights around 500–700 m, and still well within the frictional boundary layer that is at least 1 km deep or more (e.g., Kepert 2006, Fig. 6; Montgomery et al. 2006, Fig. 4; Sanger et al. 2014, Fig. 10; Montgomery et al. 2014, Figs. 8–10; Zhang et al. 2011). In view of these observations, is the height of maximum tangential wind a defensible definition for the boundary layer height in the model? What would happen if a more realistic boundary layer height were incorporated into the theory?
  • Strictly speaking, the extra nonlinear term on the right-hand side of Eq. (8), involving in part the azimuthal vorticity and vertical velocity at the radius of maximum tangential wind, arises from the inclusion of the nonlinear momentum advection terms above the boundary layer where frictional and diffusive effects are everywhere neglected (see BR09, p. 3058). Apparently, the putative “supergradient contribution” does not include the agradient effects in the frictional boundary layer, which are responsible for the boundary layer inflow in the first place. Since the Bernoulli-like function used to integrate along streamlines on either side of the tangential wind maximum in the eyewall of the vortex does not include friction, the azimuthal vorticity and the vertical velocity that are required to close the theory at the radius of maximum tangential wind must be determined separately as part of a full boundary layer calculation. In fact, the numerical models in this study are used to supply these values at the radius of maximum tangential wind, close to where the air decelerates rapidly and turns upward (Montgomery and Smith 2017, p. 567).3 In addition, like the gradient wind PI, the radius of maximum tangential wind is not predicted by the theory.
  • Further highlighting the foregoing concerns, the second term on the right-hand side of Eq. (8) cannot be evaluated from environmental soundings, but rather must be computed from the hurricane solution, itself, which is unknown a priori. RE19 acknowledge this feature, but given the inability of the formulation to predict the radius of maximum tangential wind, the radius of maximum updraft, the maximum updraft velocity, or the wind structure outside of the eyewall, is it really proper to refer to this formulation as “accounting for the supergradient contribution” if an important part of this contribution requires a determination of the boundary layer flow and the radius of maximum tangential velocity?
  • Whereas the BR09 formulation for PIa [Eq. (8)] has certainly proven to be a significant improvement over PIg [Eq. (7)] for axisymmetric hurricane simulations in the limit of small horizontal mixing length (e.g., BR09; Schecter 2013), the foregoing considerations would suggest that the reader would benefit by knowing that the BR09 formulation is only a provisional upper bound, and more accurate upper bounds might be possible incorporating the fully nonlinear boundary layer effects noted above. Indeed, technically speaking, one should not speak of a bound unless a theory is complete. At the very least, we think there should be justification for the neglect of the full radial momentum equation used to obtain Eq. (8) (cf. Smith et al. 2008; Smith and Montgomery 2008).4
  • Finally, the BR09 formulation that yields Eq. (8) assumes strictly pseudoadiabatic thermodynamics, where all condensed water immediately precipitates to the surface, and where the effects of water loading on rising moist parcels and evaporative cooling in precipitating regions are neglected. It is not entirely clear to us whether reversible or irreversible thermodynamics is used in the gradient wind PI or surface PI (to be discussed below). [For example, Eq. (9) is stated as assuming reversible moist entropy, but the last term in this equation is later stated to be “an irreversible source of entropy.” This term is then carried through the derivation and asserted to be small after Eq. (13).] If a largely reversible formulation was used, then it would seem somewhat inconsistent to compare formulations based on reversible thermodynamics against a pseudoadiabatic formulation.

5. Surface PI

In section 2 of their paper, RE19 derive a “new form of potential intensity bounding the maximum magnitude of the surface winds” using the idea of a “differential Carnot cycle.” They state (p. 1707) that the differential Carnot cycle formulation has the advantage of “only requiring the Carnot cycle’s assumptions to be valid for the part of the secondary circulation located in the eyewall of the TCs (tropical cyclones, our addition), which is easier to satisfy.” As noted by RE19, Hakim (2011)5 showed that the “secondary circulation of a simulated axisymmetric TC corresponds approximately to a Carnot cycle in the inflow and in the eyewall, but not in the outflow and subsidence regions.” The purported advantage of the differential Carnot formulation is that it is a way to avoid the explicit assumptions required to close the cycle in the outer part of the vortex. If correct, it would seem to suggest a novel way to analyze thermodynamic cycles and related heat engines in some applications without having to explicitly formulate more than half of the cycle! However, we have questions about the integral calculus used in this new formulation and these lead us to question the integrity of the cartoon on which the formulation is based (see section 6).

In their derivation of the expression for the maximum surface tangential wind [PIs; their Eq. (16)], RE19 make several physical and mathematical steps that are puzzling to us:

  • From a mathematical perspective, the differential Carnot cycle model is based on the supposition that upon integrating a combination of Bernoulli-like and heating (entropy)-rate functions around two very similar closed paths in the meridional plane (A–B–C–D–A and A–B′–C′–D–A; see their Fig. 1), the difference of these integrals will, in the limit as these two loops approach one another,6 yield a finite result in the form of an expression for the square of the maximum surface tangential velocity. At first blush, this would seem impossible, since, in the limit, the loop integrals must coincide. That is, the difference between these two closed line integrals in the meridional plane must approach zero. The only possible way out of this conundrum is if the integrands possess a singularity (an infinity, or equivalently, a Dirac delta function) somewhere along the residual loop (B′–B–C–C′–B′) before the limit is taken. However, all physical variables employed in the formulation are finite and possess no stated singular structure. How, then, is it possible to obtain a finite result from this construction?
  • Elaborating the previous point, the derivations on p. 1701 would have been easier to follow had the authors written down the variable of integration.7 In particular, it is hard to see how Eq. (15) is obtained from Eq. (14), since it would appear that the integrals on each side of the equation have been effectively canceled and replaced by a point evaluation of the integrands.
  • In the description of the Carnot cycle, RE19 state that the absolute angular momentum M that is lost to surface friction in the inflowing leg is regained in the segments C–D and C′–D through “irreversible mixing” (p. 1700, right column). Bearing in mind that M is not diffused radially, but rather the angular velocity ω = υ/r, how does irreversible mixing restore the angular momentum lost and why is this mixing not invoked until the parcel reaches a suitably large radius?
  • In their cartoon, angular momentum is lost by the surface frictional torque between points B′ and B. In a steady flow, the same amount of angular momentum must be replenished in the layer going from C to C′. It is unclear physically why the energy required for this M replenishment would be negligible compared to the heat energy used to offset the loss of M between B′ and B. However, in the derivation of Eq. (15), the upper-level energy contribution is neglected compared to the near-surface contribution. The justification for this step is not clear.
  • In the leg C–D and C′–D, the air is assumed to descend isothermally. However, we were under the impression that the isothermal outflow assumption was “poor” (Emanuel 2012, p. 989).8 This formulation seems to be a return to the original Carnot formulation of E86 that the second author of RE19 has recently spent effort to improve upon. Similarly, in the upper portion of legs D–A and D′–A, the air parcels are assumed (p. 1700, right column) to lose their moist entropy by radiative cooling, but regain it “through irreversible mixing” as they approach A. A similar question raised about angular momentum can be asked again here for the moist entropy.
  • As a prelude to the derivation, RE19 (p. 1701) note that they neglect the ice phase, for the sole reason that by “including it would add terms related to thermodynamically irreversible ice-phase effects such as supercooling.” That would not seem a sufficient reason from a physical point of view and the reader is left wondering how this neglect can be justified, given that for much of an air parcel’s ascent along the path B–C (or B′–C′), the parcel temperature would be below freezing.
  • In going from Eq. (14) to Eq. (15), the outflow temperature and Carnot efficiency factor has appeared. However, in the integration path along the near-surface leg B′–B there is no contact with the outflow layer.9 It is thus unclear how this combination of terms appears “using classical aerodynamic flux formulas for the sea surface source of enthalpy and sink of momentum” (and entropy production associated with dissipative heating) along this path.

6. A revised cartoon for surface PI

Some of the foregoing issues may be related to the cartoon of RE19’s Fig. 1, which, for a steady-flow configuration, is not consistent with the continuity equation. To satisfy continuity, the dashed (inner) contour should be extended to a complete loop that lies inside the outer loop (A–B–C–D–A). In essence, for a steady flow that satisfies continuity, it is not possible for air parcels moving along the inner (dashed) contour to extend radially as far as C′, but rather they must descend at a slightly lesser radius. Likewise, it is not possible for an air parcel to move from B′ to B as indicated. This is because air parcels rising at B must flow inward along a trajectory A–B that is slightly lower than the trajectory of air parcels rising at B′. In essence, air parcels rising at B′ cannot rise from the trajectory A–B.10 These considerations require a revision of RE19’s Fig. 1 as shown in Fig. 1 above. To fulfill the continuity constraint, two neighboring contours traversing the entire Carnot cycle are required, with an inner loop indicated by the circuit A′–B′–C′–D′–A′. The physical processes envisioned for the inner loop would be assumed to operate in a similar way for the outer loop.

Fig. 1.
Fig. 1.

Revised cartoon of the proposed differential Carnot cycle that respects the continuity equation. The yellow region depicts the primary eyewall cloud and upper-level cirrus cloud. See text for details.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0175.1

The inclusion of a second loop would imply a negative radial gradient of M between the inner and outer loop at upper levels. This negative gradient implies the existence of centrifugal (inertial) instability and nonlinear mixing in this region that would tend to neutralize the negative gradient. Nevertheless, it is unclear to us how angular momentum can be restored in this configuration to maintain the envisaged overturning circulation in a steady state. In this regard it should be remembered that, as noted above, angular momentum is not diffused radially per se.

The requirement for two loops would appear to thwart the possibility of constructing a differential Carnot cycle as a means to avoid considering the explicit processes that operate between the subsiding and inflow portions of the loop, C–D–A–B and C′–D′–A′–B′ in our Fig. 1. Moreover, it would seem peculiar to derive a PI formula solely from the changes between B′ and B when air parcels do not go from B′ to B. It is unclear to us that taking the limit as the two contours become closer together would resolve this problem. Indeed, given the need for two contours, it is unclear what any limit would mean.

7. Nonclosure of surface PI

In their conclusions, RE19 state that “While PIa applies to an actual wind speed and is very useful in assessing the contribution of supergradient flow to azimuthal winds, its computation relies on dynamical diagnostics. PIs on the other hand, is a straightforward thermodynamic bound on surface winds, a quantity that is more relevant to hurricane risk assessment.” We are puzzled by this remark since the formula for PIs [Eq. (16)] depends on a knowledge of ks*, k10, and Tout, none of which are known a priori, but must be determined by running a numerical model (see their section 3b). Indeed, as can be seen in their Fig. 5, PIs depends rather strongly on the assumed horizontal mixing length for heat and momentum, which are presumably taken to be the same in the numerical model used. The same remarks would seem to apply to PIg. If this is the case, in what sense is the bound for PIs straightforward that makes it more useful than PIa, for example? In other words, it would seem that, unlike the PI formulation of E86 and Emanuel (1995), none of the PI’s discussed in this paper represent closed theories.

8. Concluding remarks

Here we have sought to articulate questions and concerns that arose while studying RE19’s presentation of their new largely reversible, axisymmetric, PI theories, as well as a prior pseudoadiabatic, axisymmetric, formulation, which retains full nonlinearity above the boundary layer. Notwithstanding their efforts to verify these theories using two axisymmetric numerical models, without convincing answers to these questions, we remain skeptical about the integrity of the new PI theories, especially that for the maximum tangential wind near the surface. For one thing, we see such theories to be of limited utility if one has to run a numerical model in order to calculate them. However, a more serious issue we have is that the differential Carnot cycle underpinning the theory does not respect the continuity constraint on the flow. Attempts to remedy this problem casts doubt on the ability to invoke the differential Carnot cycle framework.

Acknowledgments

MTM acknowledges the support of NSF Grant AGS-1313948, IAA-1656075, ONR Grant N0001417WX00336, and the U.S. Naval Postgraduate School. The views expressed herein are those of the authors and do not represent sponsoring agencies or institutions.

REFERENCES

  • Bell, M. M., and M. T. Montgomery, 2008: Observed structure, evolution, and potential intensity of category 5 Hurricane Isabel (2003) from 12 to 14 September. Mon. Wea. Rev., 65, 20252046, https://doi.org/10.1175/2007MWR1858.1.

    • Search Google Scholar
    • Export Citation
  • Bryan, G. H., and R. Rotunno, 2009: Evaluation of an analytical model for the maximum intensity of tropical cyclones. J. Atmos. Sci., 66, 30423060, https://doi.org/10.1175/2009JAS3038.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Emanuel, K. A., 1986: An air–sea interaction theory for tropical cyclones. Part I: Steady-state maintenance. J. Atmos. Sci., 43, 585604, https://doi.org/10.1175/1520-0469(1986)043<0585:AASITF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Emanuel, K. A., 1995: Sensitivity of tropical cyclones to surface exchange coefficients and a revised steady-state model incorporating eye dynamics. J. Atmos. Sci., 52, 39693976, https://doi.org/10.1175/1520-0469(1995)052<3969:SOTCTS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Emanuel, K. A., 2012: Self-stratification of tropical cyclone outflow. Part II: Implications for storm intensification. J. Atmos. Sci., 69, 988996, https://doi.org/10.1175/JAS-D-11-0177.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Emanuel, K. A., and R. Rotunno, 2011: Self-stratification of tropical cyclone outflow. Part I: Implications for storm structure. J. Atmos. Sci., 68, 22362249, https://doi.org/10.1175/JAS-D-10-05024.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hakim, G. J., 2011: The mean state of axisymmetric hurricanes in statistical equilibrium. J. Atmos. Sci., 68, 13641376, https://doi.org/10.1175/2010JAS3644.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kepert, J. D., 2006: Observed boundary-layer wind structure and balance in the hurricane core. Part II. Hurricane Mitch. J. Atmos. Sci., 63, 21942211, https://doi.org/10.1175/JAS3746.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Montgomery, M. T., and R. K. Smith, 2017: Recent developments in the fluid dynamics of tropical cyclones. Annu. Rev. Fluid Mech., 49, 541574, https://doi.org/10.1146/annurev-fluid-010816-060022.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 13351347, https://doi.org/10.1175/BAMS-87-10-1335.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Montgomery, M. T., J. A. Zhang, and R. K. Smith, 2014: An analysis of the observed low-level structure of rapidly intensifying and mature Hurricane Earl (2010). Quart. J. Roy. Meteor. Soc., 140, 21322146, https://doi.org/10.1002/qj.2283.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Persing, J., M. T. Montgomery, J. McWilliams, and R. K. Smith, 2013: Asymmetric and axisymmetric dynamics of tropical cyclones. Atmos. Chem. Phys., 13, 12 29912 341, https://doi.org/10.5194/acp-13-12299-2013.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Rousseau-Rizzi, R., and K. Emanuel, 2019: An evaluation of hurricane superintensity in axisymmetric numerical models. J. Atmos. Sci., 76, 16971708, https://doi.org/10.1175/JAS-D-18-0238.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sanger, N. T., M. T. Montgomery, R. K. Smith, and M. M. Bell, 2014: An observational study of tropical cyclone spinup in Supertyphoon Jangmi from 24 to 27 September. Mon. Wea. Rev., 142, 328, https://doi.org/10.1175/MWR-D-12-00306.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Schecter, D. A., 2013: Relationships between convective asymmetry, imbalance and intensity in numerically simulated tropical cyclones. Tellus, 65A, 20168, https://doi.org/10.3402/tellusa.v65i0.20168.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Smith, R. K., and M. T. Montgomery, 2008: Balanced boundary layers used in hurricane models. Quart. J. Roy. Meteor. Soc., 134, 13851395, https://doi.org/10.1002/qj.296.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Smith, R. K., M. T. Montgomery, and S. Vogl, 2008: A critique of Emanuel’s hurricane model and potential intensity theory. Quart. J. Roy. Meteor. Soc., 134, 551561, https://doi.org/10.1002/qj.241.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zhang, J. A., R. F. Rogers, D. S. Nolan, and F. D. Marks, 2011: On the characteristic height scales of the hurricane boundary layer. Mon. Wea. Rev., 139, 25232535, https://doi.org/10.1175/MWR-D-10-05017.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
1

This is a reminder that three-dimensional hurricane simulations and observations are the proper benchmark for maximum intensity formulations and that strictly axisymmetric simulations, with their intrinsically axisymmetric rings of convection, have fundamental limitations (Persing et al. 2013).

2

If the flow within the boundary layer differs from Vg,b, would not one then need a separate equation for this velocity? Our interpretation is consistent with that of BR09 (p. 3045), who, in their derivation of gradient wind PI (and its nonlinear generalization discussed in the forthcoming section 3), explicitly assume that the boundary layer is well mixed: “The derivation considers the variables s, M, and radial velocity (u) to be well mixed (i.e., constant) in the PBL.” Here, BR09 are following E86 (p. 593).

3

In Eq. (8), the direct contribution from the boundary layer enters through the left-hand-side term and first right-hand-side term via implementation of the E86 boundary layer closure [Eq. (3)] (see also BR09, p. 3054), together with the assumption that the boundary layer flow is well mixed in absolute angular momentum (hence tangential velocity) and moist entropy sb at leading order.

4

Smith and Montgomery (2008) show, in fact, that this approximation cannot be justified on the basis of a simple scale analysis of the vortex boundary layer equations.

5

Noted also by J. Persing (2002, personal communication).

6

“Taking the circuit B′–B–C–C′–B′ to be of infinitesimal width” (p. 1701).

7

Technically speaking, in the absence of the differential line increment, the residual sums being manipulated are individually infinite.

8

Emanuel and Rotunno (2011) demonstrated that in numerically simulated tropical cyclones, the assumption of constant outflow temperature is poor and that, in the simulations, the outflow temperature increases rapidly with angular momentum.”

9

The contributions in Eq. (14) along the path B–C–C′–B′ are assumed to be negligible (and hence zero), leaving just the path integrals along the near surface.

10

In contrast, this fact is properly accounted for in Fig. 11 of BR09.

Save