Hurricane Maximum Intensity: Past and Present

1704 MONTHLY WEATHER REVIEW VOLUME 129

J. PARKS CAMP National Weather Service, Sterling, Virginia

MICHAEL T. M ONTGOMERY Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado

(Manuscript received 6 March 2000, in ®nal form 15 September 2000)

ABSTRACT Hurricane intensity forecasting has lagged far behind the forecasting of hurricane track. In an effort to improve the understanding of the hurricane intensity dilemma, several attempts have been made to compute an upper bound on the intensity of tropical cyclones. This paper investigates the strides made into determining the maximum intensity of hurricanes. Concentrating on the most recent attempts to understand the maximum intensity problem, the theories of Holland and Emanuel are reviewed with the objective of assessing their validity in real tropical cyclones. Each theory is then tested using both observations and the axisymmetric hurricane numerical models of Ooyama and Emanuel. It is found that ambient convective instability plays a minor role in the determination of the maximum intensity and that the Emanuel model is the closest to providing a useful calculation of maximum intensity. Several shortcomings are revealed in Emanuel's theory, however, showing the need for more basic research on the axisymmetric and asymmetric dynamics of hurricanes. As an illustration of the importance of asymmetric vorticity dynamics in the determination of a hurricane's maximum intensity it is shown, using Ooyama's hurricane model, that the maximum intensity of a tropical cyclone may be diminished by convectively generated vorticity anomolies excited outside the primary eyewall. The vorticity anomolies are parameterized by adding a concentric ring of vorticity outside the primary eyewall that acts to cut off its supply of angular momentum and moist enthalpy. It is suggested that the generation of vorticity rings (or bands) outside the primary eyewall is a major reason why tropical cyclones fail to attain their maximum intensity even in an otherwise favorable environment. The upshot of this work points to the need for obtaining a more complete understanding of asymmetric vorticity processes in hurricanes and their coupling to the boundary layer and convection.

1. Introduction intensity problem, numerous attempts have been made to compute an upper bound on tropical cyclone intensity Tropical cyclones plague tropical and subtropical for given atmospheric and oceanic conditions. For the oceans as well as neighboring land areas around the purposes of the present study, the maximum intensity world on an annual basis. Research into understanding of a hurricane [hereafter called maximum potential in- these deadly vortices has been ongoing for many years. tensity (MPI)] is de®ned as the maximum intensity (as Much progress has been made into understanding some determined by minimum surface pressure or maximum of the basic mechanisms that govern tropical cyclones, tangential winds) that a tropical cyclone may achieve allowing for signi®cant strides to be made in the fore- for a given atmospheric and oceanic thermal structure. casting of track. However, the associated forecasts of Generally, we assume that upper-level winds are fa- tropical cyclone intensity have not shown the same im- vorable for development. Investigations into MPI have provement (DeMaria and Kaplan 1999 and references). begun from different starting points leading to differing Understanding what controls hurricane intensity is vital viewpoints on the structure of tropical cyclones. The in order to properly warn those in the path of storms, primary goals of this paper are twofold. The ®rst goal as well as in predicting the impacts of global climate is to evaluate the current state of MPI theory and de- change on the character of tropical cyclones (Knutson termine which proposed theory best correlates with the and Tuleya 1999 and references). structure and behavior of observed tropical cyclones. As a step toward understanding the tropical cyclone The second is to suggest an answer to why most tropical cyclones fail to reach their MPI. Corresponding author address: J. Parks Camp, National Weather Kleinschmidt (1951), Miller (1958), and Malkus and Service, 44087 Weather Service Rd., Sterling, VA 20166. Reihl (1960) conducted the ®rst such studies aimed at E-mail: [email protected] determining the upper limit of tropical cyclone intensity.

᭧ 2001 American Meteorological Society

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uel decreases with decreasing relative humidity such that a surface relative humidity of 100% at the eyewall does not support any circulation whatsoever. Converse- ly, the strongest storms in Holland's theory occur with 100% surface relative humidity under the eyewall, with only weak circulations (tropical depression strength) supported at 75% RH. This result is striking and suggests that the two models are fundamentally different. As we shall see, the hypothesized role of air±sea energy exchange and convective available potential energy (CAPE) lies at the heart of this difference. The outline of this paper is as follows. We ®rst review the basis of the MPI theories of Holland (1997) and Emanuel [1986, 1988a,b, 1989, 1991, 1995a,b, 1997, FIG. 1. Dependence of minimum sea level pressure (MSLP) on collectively referred to hereafter as Emanuel (1986± eyewall surface relative humidity (see text for details) predicted by 97)]. This is followed by a discussion of shortcomings Holland's (1997) (solid) and Emanuel's (1997) (dashed) MPI theories. present in each model, including a comparison of Eman- Initial conditions: T ϭ 300 K and p ϭ 1015 mb. s env uel's theory with the numerical models of Emanuel (1995b) and Rotunno and Emanuel (1987). The adverse Kleinschmidt (1951) modeled the tropical cyclone with effects of secondary eyewalls on tropical cyclone in- a frictional boundary layer beneath a conditionally neu- tensity are examined next using Ooyama's (1969) hur- tral out¯ow layer very similar, in many respects, to the ricane model. It is suggested that secondary eyewalls recent work of Emanuel (1986) [see Gray (1994) for a and the convectively generated vorticity anomalies that comparison of these two approaches]. Malkus and Reihl spawn them are the principle reason why most hurri- (1960) examined parcel trajectories in the in¯ow layer canes fail to reach their MPI even under favorable en- of tropical cyclones, while Miller (1958) developed an vironmental conditions based on SST, out¯ow temper- MPI theory controlled by sea surface temperature (SST) ature, vertical shear, etc. Finally, the main ®ndings are and the height of the convective equilibrium level (EL). summarized and paths for future research are discussed. Camp (1999) provides a detailed review of each of these approaches. 2. Holland (1997) In recent years, the most widely recognized investigations of MPI are those of Emanuel (1986, 1988a,b, Nearly 40 years after Miller introduced his MPI the- 1991, 1995a, 1997) and Holland (1997). While the ory, Holland (1997) introduced a similar, yet revised method used by Holland is similar to that of Miller, MPI theory. As with Miller (1958), Holland's theory Emanuel takes a very different approach leading to a relies heavily on the presence of ambient CAPE to de- revised view of tropical cyclone thermodynamic struc- termine the MPI. Both theories assume that surface air ture. The MPI in both models is governed by SST and rises moist adiabatically in the eyewall before sinking surface relative humidity. Additionally, each MPI for- dry adiabatically (with mixing from the eyewall) within mulation is also a function of the thermal structure of the eye. The surface pressure fall due to the warming the upper troposphere, with Holland's MPI regulated by from moist-adiabatic ascent is termed the ``one cell'' the EL, and Emanuel's MPI regulated by the average theory. The surface pressure fall due to the warming temperature of the out¯ow region. Figure 1 shows the from the adiabatic descent of parcels following moist MPI as a function of surface relative humidity at the adiabatic ascent is termed the ``two cell'' theory. eyewall calculated by Holland's and Emanuel's meth- Miller's (1958) and Holland's (1997) approaches are ods. The relative humidity in Holland's model is spec- modi®cations of the one and two-cell theories. The pri- i®ed underneath the eyewall, presumably at the radius mary change made by Holland (1997) to Miller's (1958) of maximum updraft, while the relative humidity in the approach is to utilize the pressure dependence of moist Emanuel model is strictly valid at the radius of maxi- entropy [proxied by equivalent potential temperature mum winds. The calculations performed were made us- (␪e)], so that as convection warms the eyewall and low- ing unaltered code generously provided by K. Emanuel ers the surface pressure, the boundary layer ␪e increases and G. Holland, respectively. With similar initial con- as well, allowing for a further warming in the eyewall ditions, a sea surface temperature of 300 K, and a sur- and a further pressure drop at the surface. For most face relative humidity of 86%, the two theories predict initial conditions, the pressure drops under the eyewall approximately the same MPI. However, as is shown by become smaller with each iteration, leading to a con- Fig. 1, this agreement is purely coincidental. While both vergence of the surface pressure under the eyewall. theories are sensitive to the surface relative humidity Once the eyewall surface pressure converges, an eye near the eyewall radius, they are sensitive in opposite parameterization is used if the net pressure fall under ways. The minimum sea level pressure given by Eman- the eyewall is more than 20 mb. It is argued that a system

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FIG. 3. Minimum sea level pressure from Miller's (1958) model (solid), and Holland's (1997) model (dot), as a function of sea surface

temperature with RHs ϭ 85%. Results from one-cell (dash) and two- FIG. 2. Skew T±Logp thermodynamic diagram showing Jordan cell (dash±dot) trajectories are also shown. (1958) mean ``hurricane season'' sounding (solid line), and associated Miller-type (dash) and Holland-type (dash±dot) synthesized eye soundings for Ts ϭ 300 K and RHs ϭ 85%. 301 K. Above this temperature, however, the surface pressures predicted by Holland decrease rapidly, falling well below the values from even the two-cell model. with a pressure fall less than 20 mb is unlikely to have Holland's surface pressures range from 960 mb at Ts ϭ the structure of a mature hurricane, thus there would be 299 K to an incredibly low pressure of 770 mb at Ts ϭ no eye. For such weak systems, the surface pressure 304 K. For a surface temperature of 305 K, the Holland calculated from the eyewall iteration is the MPI for the model does not reach convergence. The surface pressure associated environment. A pressure drop under the eye- achieved with a surface temperature of 304 K is over wall of greater than 20 mb initiates the construction of 100 mb lower than the corresponding surface pressure an eye sounding. The dynamical processes responsible from the Miller model. For temperatures of 302 K and for determining the thermodynamic structure of the eye higher, Holland predicts surface pressures below those are not considered. Only the structure of the ®nal state of the strongest tropical cyclones ever observed, even is evaluated. To determine this structure, the equivalent though sea surface temperatures of 302 K and greater potential temperature throughout the eye is assumed are common in tropical oceans. Since the eye parame- equal to the surface saturation equivalent potential tem- terization does not differ substantially between the Mill- perature under the eyewall once the iteration process is er and Holland models, the large difference in MPI must complete. Compared to the empirical eye of Miller be a result of the eyewall iteration. This procedure has (1958) (dashed line in Fig. 2), Holland's parameteri- a small impact at lower surface temperatures, when the zation yields an eye structure that is in better agreement initial pressure drop is small and the procedure con- with observations (Hawkins and Rubsam 1968). The verges rapidly, but has an enormous impact at higher Holland eye, shown in Fig. 2 (dash±dot), generates a surface temperatures, when the initial pressure drop is maximum temperature anomaly between 300 and 400 large and the eyewall iteration converges slowly. The mb, higher than the 500±600-mb anomaly found using Miller and Holland model therefore give similar results Miller. for weaker systems, but dramatically different results Maximum potential intensity predictions using Hol- for stronger systems. land's model are shown by the dotted line in Fig. 3. The low pressures predicted by the Holland model Analogous predictions from Miller's (1958) approach bring to mind the hypercanes predicted by Emanuel as well as the straight one- and two-cell approaches are (1988a) and Emanuel et al. (1995) (see section 3). In shown for comparison. The relative humidity under the the present formulation, however, the extremely low sur- eyewall is assumed to be 90%, following Holland face pressures appear to be a product of raising the (1997), and the Jordan (1958) mean ``hurricane season'' surface temperature while keeping the remainder of the sounding is used for the environment. Recall that the sounding the same. This has the effect of producing a Holland model has little dependence on the relative hu- large amount of ambient CAPE, which leads to very midity of environmental air since the air parcels are large pressure falls under the eyewall. If a convectively assumed to reach a relative humidity of 90% by the time neutral sounding with a surface temperature of 308 K, they reach the eyewall. While the methods used by Hol- ambient surface relative humidity of 80%, ambient sur- land and Miller are closely related, their predictions are face pressure of 1015 mb, and a tropopause at 100 mb dramatically different. The MPI predicted by each mod- is constructed, the Holland model predicts a minimum el is in fair agreement for surfaces temperature below sea level pressure of only 907 mb. This predicted pres-

Unauthenticated | Downloaded 09/25/21 08:17 AM UTC JULY 2001 CAMP AND MONTGOMERY 1707 sure is higher than the lowest pressures observed in the intense hurricanes such as Hurricane Gilbert (1988) most intense tropical cyclones forming over oceans with (Dodge et al. 1999). Since the hydrostatic surface pres- temperatures less than 305 K. It then becomes clear how sure under the eyewall is determined by the mass of the important an initially unstable environment is in pro- atmosphere directly above, such a vertical pro®le for a ducing tropical cyclones with very low central pressures sloping eyewall would extend through the lower portion in Holland's model. This is in sharp contrast to the of the eyewall and then through the outer edge of the Emanuel model, which, for low enough out¯ow tem- eye itself. Since the warmest temperature anomalies are peratures, high enough sea surface temperatures, or a found in the eye, the associated surface pressure under combination of both, predicts incredibly intense cy- the sloped eyewall would be lower than the analogous clones (hypercanes) in an initially neutral environment.1 surface pressure under a vertical eyewall. Consequently, The unrealistically low surfaces pressures predicted a more realistic eyewall structure would tend to lower by Holland (1997) raise concerns about the general va- the minimum central pressure predicted by Holland even lidity of such an MPI formulation. An MPI of 800 mb more. is not useful in trying to assess the maximum intensity Miller's and Holland's theories rely on a parameter- that a developing tropical cyclone may achieve. The ized eye sounding to attain the minimum pressure. The assumptions used in the Holland model and the Miller Miller eye was arbitrarily set to match observed eye model are now examined to assess their validity and temperature and moisture pro®les. As we have already applicability to real tropical cyclones. indicated, however, the level of maximum warm anom- The sensitivity to surface relative humidity in Hol- aly produced with this particular method is well below land's theory is similar to but not as dramatic as the the level of maximum warm anomaly observed in actual sensitivity to surface temperature. The minimum surface tropical cyclones. The maximum warm anomaly from pressure decreases with increasing relative humidity, as Holland's parameterized eye is in better agreement with would be expected from a CAPE-based model since observations. But, since dynamics are not used to de- increased surface moisture yields a higher ␪e, thus a termine the eye sounding for either model, neither is warmer column, and a lower surface pressure. The MPI able to resolve the characteristic temperature inversion ranges from 884 mb at 100% relative humidity to 1002 in observed tropical cyclone eyes (Willoughby 1998). mb at 75% relative humidity for a surface temperature Whether or not the low-level inversion is important in of 300 K (Fig. 1). The jump to lower surface pressures determining the surface pressure is not yet clear, but the at relative humidities above 77% in the Holland model Holland eye in Fig. 2 bears little resemblance to the eye is due the eye parameterization being ``switched on.'' pro®les from several intense tropical cyclones reported The choice of 90% relative humidity under the eyewall by Willoughby (1998), especially below 500 mb. A con- in Holland (1997) is, in our opinion, an arbitrary choice stant ␪e in the eye, used by Holland, is not observed in for a quantity that has a large impact on the ®nal results any of the eye pro®les from Willoughby (1998). Without and for which observed values are scattered over a rel- a clear understanding of the processes that control the atively large range of values. Holland (1997) cites rel- thermodynamic structure of the eye, we think it is not ative humidities under the eyewall ranging from 80% insightful to apply an arbitrary temperature pro®le to to 95%, which corresponds to a range of MPI of 971± obtain a surface pressure, since there are an in®nite 908 mb. Thus, the uncertainty in the eyewall relative number of such pro®les that would yield the same sur- humidity leads to estimates of MPI ranging from a mod- face pressure. Conversely, there are an in®nite number erate category 1 to a devastating category 5 hurricane of pro®les that would yield a different surface pressure. on the Saf®r±Simpson scale. Without a ®rm grasp of There is no dynamical basis for assuming the chosen the moisture structure of the hurricane boundary layer, pro®le is the correct one. the MPI model of Holland (1997) appears to be of little The Holland model has an indirect reliance on the use. ocean. The dependence of surface ␪e on surface pressure In order for the surface pressure to be calculated under is the primary source of this reliance. Without any ad- the eyewall in Holland (1997), the moist-adiabatic pro- ditional heat from the ocean, surface air ¯owing from ®le associated with rising motion is used as the vertical the environment to the eyewall would adiabatically ex- temperature pro®le in the hydrostatic formulation. In pand and cool as it moves to a region of lower pressure. other words, the eyewall is assumed to be vertical. In Since Holland speci®es the surface temperature to be a nature, to conserve angular momentum, the eyewall constant from the environment into the eyewall, sensible slopes outward with height. This has been observed in heat must be added to in¯owing air to keep the parcels isothermal. This sensible heat is supplied by the ocean, which for practical purposes in this model can be con- 1 Note added in proof: A recent paper by Schade (2000) argues sidered an in®nite heat source. The role of the ocean in that ``the sensitivity of the MPI to the SST by Holland97 should be supplying latent heat to the tropical cyclone is less clear. interpreted as the sensitivity of a tropical cyclone to local changes of the SST under its eye'' (by ocean cooling). We suggest, however, Without the addition of any water vapor, the mixing that the sensitivity in Holland's formulation is an artifact of its over- ratio of an air parcel remains the same as it travels reliance on CAPE. isothermally down a pressure gradient and expands.

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Since the saturation mixing ratio increases along this duction in the in¯owing boundary layer. The latter for- same path, the relative humidity of air parcels traveling mulation is the focus of the present paper. isothermally toward the cyclone center would decrease. To investigate the possibility of maintaining a hur- Moisture added to the in¯ow from the ocean would ricane-like vortex without a conditionally unstable en- allow the relative humidity to be maintained or increase vironment, E86 developed a steady-state model of a on the inward path, while a loss of moisture would tropical cyclone in a conditionally neutral environment. further decrease the relative humidity. Since the envi- The model vortex is assumed axisymmetric, in hydro- ronmental relative humidity is not speci®ed in the Hol- static and gradient balance, and reversible thermody- land model, it is, strictly speaking, unknown whether namics are assumed. The key feature of the model is moisture is being added to or taken away from in¯owing that the CAPE is zero everywhere in the domain, such air to achieve the 90% relative humidity speci®ed under that the atmosphere is neutral to slantwise convection. the eyewall. Since the surface relative humidity of the The impact of the neutral constraint is that boundary ambient tropical atmosphere is generally lower than the layer air is neutrally buoyant when lifted along surfaces value of 90% used by Holland (1997) one can infer that of constant angular momentum. Using the boundary lay- a latent heat transfer from the ocean to the atmosphere er closure from Ooyama (1969), the steady-state non- must take place in the Holland model. However, since dimensional entropy balance under the tropical cyclone the eyewall relative humidity is arbitrarily speci®ed, the eyewall is given by Emanuel (1995a, henceforth E95a) as air±sea interaction may not be properly accounted for. ␹ Cץ Finally, the lack of a dependence on surface friction ␺ ϭϪ k (1 ϩ c|V|)|V|(␹* Ϫ ␹), (1) 0 2 s rCDץ in the Holland theory is troubling. Implicitly, surface friction is assumed to force the boundary layer radial where ␺ 0 is the radial-mass streamfunction at the top in¯ow, the convergence of which produces the updraft of the boundary layer and V the tangential speed. En- at the base of the eyewall. However, the negative impact tropy is represented as a temperature-weighted deviation of surface friction in spinning down the vortex does not from an environmental value, appear to be part of the formulation. As wind speeds increase, the effect of friction increases, such that its ␹ ϵ (TsoϪ T )(s Ϫ sba), (2) neglect in any theory of tropical cyclone structure and where T is the surface temperature, T is the out¯ow maintenance seems suspect. s o temperature, s the moist entropy of the subcloud layer,

and sba is the moist entropy of the ambient subcloud 3. Emanuel (1986±97) layer (Emanuel 1995b, henceforth E95b). The coef®- cients C and C are the sea±air exchange coef®cients Emanuel (1986, hereafter E86) introduced a theory k D of tropical cyclone structure and development that is for moist entropy and angular momentum, respectively, fundamentally different from Holland's (1997) theory. and the empirical constant c determines the wind de- There are two aspects of CAPE-dependent MPI theories pendence of the surface ¯uxes. The left-hand side of that E86 takes issue with. The main disagreement con- (1) represents the radial advection of entropy, which is cerns the role of CAPE in tropical cyclone development exactly balanced by the ¯ux of entropy from the sea and maintenance. While the CAPE-dependent MPI the- surface represented on the right-hand side. Note that the ories rely heavily on the presence of ambient CAPE in ¯ux of entropy from the ocean is controlled by the the tropical atmosphere, E86 asserts that, even though amount of disequilibrium between the boundary layer CAPE does exist in the tropical atmosphere, it is ques- and the sea surface. That is, a dry boundary layer allows tionable that tropical convection is able to utilize this for a greater entropy ¯ux than a moist boundary layer, source of energy due to entrainment, mixing, and other with zero entropy ¯ux occurring if the boundary layer small-scale processes. In addition to the overemphasis is saturated. on ambient CAPE in previous MPI theories, E86 argues The nondimensional angular momentum balance un- that the energy exchange between the boundary layer der the eyewall is similarly given by R2ץ -and the ocean is much stronger than suggested by the ories relying on ambient CAPE for energy production. ␺0 ϭ 2(1 ϩ c|V|)|V|rV, (3) r 2ץ This enhanced air±sea interaction has been recently termed wind-induced surface heat exchange (WISHE). where R, the potential radius, is analogous to angular Relationships governing the maximum tangential winds momentum. This equation states that the radial advec- and minimum sea level pressure, based on the WISHE tion of angular momentum is balanced by the ¯ux of mechanism, are derived in two ways. The popularized angular momentum into the ocean due to friction. In the derivation assumes that the tropical cyclone behaves as previous two equations, the surface exchange has been a Carnot heat engine (Emanuel 1986, 1988a,b, 1991, parameterized using the standard bulk aerodynamic for- 1997). This is, however, Emanuel's secondary method mulation; the justi®cation for and a discussion of the of deriving the MPI. The primary method is based on attending uncertainties in such an approach are provided a balance between frictional dissipation and energy pro- by E95a,b.

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Using thermal wind balance and the de®nition of moist entropy, Eqs. (1) and (3) can be combined to form a transcendental equation relating the tangential velocity and the moist entropy at the radius of maximum wind (RMW),

1 CC 1 V 2 1 Ϫ A kkഠ 1 Ϫ Ar 2 Ϫ ␹(1 Ϫ A) , (4) 2 CC 4 0 ΂΃DD[] where r 0 is the radius at which the winds decay to zero. In Eq. (4), T Ϫ T ␹ A ϵϩso s , (5) TRTsdsa(1 Ϫ RH ) where Rd is the gas constant for dry air and RHa is the FIG. 4. Minimum sea level pressure from Emanuel's model (solid) ambient relative humidity. Equation (4) is the MPI equa- as a function of sea surface temperature with RHenv ϭ 85% and penv tion for a tropical cyclone that has developed in a con- ϭ 1015 mb. Results from Holland's model are shown (dashed) for comparison. ditionally neutral environment.2 It can be solved for V and ␹ at the RMW. Emanuel then assumes an ``eye closure'' in which the tangential ¯ow inside the RMW parameters. During intensi®cation, pressure falls lead to is in solid-body rotation. This allows for the calculation an increase in the surface mixing ratio, thereby increas- of the minimum central pressure once the maximum ing the moist entropy and allowing further intensi®ca- tangential winds are known. We note, following E95a, tion. Under normal conditions, the pressure falls lead that the maximum tangential winds are not dependent to smaller and smaller increases in the mixing ratio, on the structure of the eye itself. The eye closure used allowing the solution to converge to a steady state. Un- in E95a is actually a revised version of that used in E86 der extreme conditions, however, the change in mixing to determine the minimum surface pressure. The orig- ratio increases more than is necessary to maintain the inal formulation resulted in a hydrostatic calculation pressure falls, leading to runaway intensi®cation. Eman- somewhat similar to Holland (1997) and Miller (1958), uel (1988a, henceforth E88a) hypothesizes that storms which was independent of the ratio of surface exchange forming under such conditions never reach an equilib- coef®cients. The revised formulation now depends on rium intensity, with central pressures that spiral toward the ratio of surface exchange coef®cients, in better zero. Such ``hypercane'' solutions to the MPI equations agreement with axisymmetric model results (E95a). Fig- occur for the combination of low out¯ow temperatures ure 4 presents MPI calculations for various sea surface and high sea surface temperatures, conditions that do temperatures and an environmental relative humidity of not exist under current climate conditions. For example 85% using Emanuel's (E95a) formulation. Results using (from E88a), conditions leading to the hypercane regime Holland's (1997) method are shown for comparison. The would require Ts ϭ 36ЊC, To ϭϪ110ЊC, and surface predicted minimum pressures from the Emanuel method relative humidity RHa ϭ 80%. Wind speeds in such are more reasonable than those from the Holland meth- storms could approach speeds of nearly 300 m sϪ1 before od. A minimum pressure of just below 900 mb at a sea internal dissipation limits the intensi®cation (Emanuel surface temperature of 305 K is in good agreement with et al. 1995). Emanuel et al. (1995) have suggested that the minimum pressures observed in intense tropical cy- the mass extinction at the end of the Cretaceous period clones. may have been, in part, due to the presence of hypercanes triggered by an asteroid/meteor impact or extreme volcanic activity. Hypercanes Limits to such hypercanes are hypothesized to exist A unique feature of the transcendental equation given by E88a, but such limits are not realized by the MPI by (4) arises for high values of surface temperature, low equations due to the assumptions that underly their der- values of out¯ow temperature, or a combination of both. ivation. E88a hypothesizes three conditions that may A point is reached, at which no steady-state solution to limit the intensity of tropical cyclones in the hypercane the equations is possible for the given environmental regime. First, a nonisothermal boundary layer would limit the amount of heat available to the secondary circulation, limiting the intensity. In fact, observations of 2 A revision to this formulation has been proposed by Bister and nonhypercane regime tropical cyclones under current Emmanuel (1998) incorporating the effect of dissipative heating in atmospheric conditions indicate that the boundary layer the boundary layer. The only impact of this revision is to change T S is not exactly isothermal (Cione et al. 2000; Shay et al. to TO in the denominator of the ®rst term in Eq. (5), leading to somewhat stronger storms. This modi®cation has been used in all of 1992). The fact that nonisothermal boundary layers exist the Emanuel MPI calculations presented here. indicate that surface ¯uxes of sensible heat from the

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After reaching the maximum intensity of 283 m sϪ1, the vortex weakens rapidly with tangential winds dropping below 100 m sϪ1 followed by a period of slower weakening. This rapid weakening is attributed, by Emanuel et al. (1995), to vortex breakdown resulting from internal dissipation, although the physics triggering the breakdown and the subsequent evolution is not yet well understood.

4. Limitations of Emanuel's MPI There are two potential caveats in Emanuel's MPI theory, the assumption of constant relative humidity from the environment to the eyewall, and the assumption of neutrality to slantwise convection in the eyewall. The FIG. 5. Time evolution of maximum tangential wind predicted by the tropical cyclone model of E95b. Initial conditions: T ϭ 307 K moist neutrality assumption is only strictly valid in the s ambient environment and in the eyewall when the trop- and penv ϭ 1015 mb, To ϭ 171 K. ical cyclone is in a steady state. While the cyclone is intensifying, the moist neutral assumption is not exactly ocean are unable to fully keep up with adiabatic cooling satis®ed, since as ␪e increases at the surface, small even for tropical cyclones of present-day intensity, much amounts of CAPE are generated, leading to some pos- less for tropical cyclones with tangential velocities of itive buoyancy in the eyewall. In the axisymmetric the- 300msϪ1 or more. A second limit may occur due to ory, some buoyancy is necessary to warm the vertical strong internal dissipation (in the eyewall), which be- column, as parcels moving neutrally through the eyewall comes important at high wind speeds, and therefore the would not produce any warming. Thus, Emanuel is not assumptions that all dissipation occurs only in the suggesting that the tropical cyclone is entirely absent of boundary layer and at large radii in the out¯ow break convective instability, just that any buoyancy that de- down. Finally, the calculation of an out¯ow temperature velops during the intensi®cation process is quickly ex- from the environmental temperature pro®le may not be tinguished as the eyewall warms. valid in the case of the hypercane, where out¯ow is so The steady-state assumption of neutrality to slantwise strong (supercritical or nearly so) that the hurricane vor- convection also deserves some discussion. Such an as- tex itself controls the out¯ow temperature. None of the sumption implies that air moving along surfaces of con- previous arguments are included in the derivation of the stant angular momentum would not experience any up- MPI equations; thus, they may conceivably place strong ward acceleration. In fact, as air parcels move up constraints upon the maximum intensity of hypercanes, through the eyewall, and outward into the out¯ow, their or prohibit them altogether. vertical velocity should decrease owing to the smaller Emanuel (1989, 1995b) developed a simple axisym- and smaller vertical component of motion. Additionally, metric numerical tropical cyclone model based on the the radial velocity would be outward throughout the assumption of neutrality to slantwise moist convection. out¯ow layer. Without any inward component of ve- The model was obtained for the present study from K. locity in the midlevels, there can be no inward advection Emanuel and several experiments using it have been of high angular momentum air. This raises the question conducted. Details of the model formulation can be of how the steady-state vortex maintains itself against found in Emanuel (1989, 1995b). For the present dis- frictional spindown without midlevel in¯ow. That a vor- cussion, Emanuel's (1995a) numerical model was run tex can be maintained in a steady state without signif- for a particular case where the MPI equations do not icant midlevel in¯ow above the boundary layer is sup- converge to a solution. The input parameters are the sea ported by the hydrostatic model results of Emanuel surface temperature (Ts ϭ 307 K) and the out¯ow tem- (1989, 1995a). Figure 6 shows the r±z distribution of perature (To ϭ 171 K). These values were chosen by the tangential velocity ®eld for a steady-state vortex examining Figs. 3 and 4 of E88 and choosing values calculated from the Emanuel's (1995a) hydrostatic mod- that were well into the hypercane regime. All other model with maximum tangential winds of 76 m sϪ1. Figure el parameters were kept at the default values. The time 7 shows the corresponding ®elds of radial velocity (pan- evolution of the maximum tangential wind at the top of el a) and vertical velocity (panel b). Figure 8 shows the the boundary layer predicted by Emanuel's numerical two-dimensional vector ®eld of the transverse circula- model is shown in Fig. 5. The maximum tangential ve- tion corresponding to Fig. 7. Figure 7a clearly shows locity reaches 283 m sϪ1 at approximately t ϭ 20 days. that all of the in¯ow is con®ned to the lowest 1.75 km For the given surface temperature, this is about 80% of of the model atmosphere, with out¯ow dominant in the the speed of sound (ϳ351msϪ1). The vortex appears middle and upper troposphere. Inside a radius of 50 km, to intensify in stages on its way to its peak intensity. no in¯ow occurs above 1.75 km. It appears then that a

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FIG. 8. Vector ®eld of transverse circulation from hydrostatic model FIG. 6. Steady-state two-dimensional tangential velocity ®eld of Emanuel (1995a). Note that vectors are stretched in the vertical (m sϪ1; contour interval,5msϪ1) from hydrostatic model of Emanuel direction so that the vertical component of the vectors is exaggerated. (1995a). suf®cient amount of angular momentum is being ad- curring within the updraft, which is expected with an vected inward near the top of the boundary layer to accelerating updraft. It appears then that some buoyancy maintain the tropical cyclone against friction. Such a in the eyewall is present in the steady state. While it is structure with highly concentrated in¯ow is not nec- true that the isolines of angular momentum and moist essarily supported by observations. Analysis of the sec- entropy are nearly congruent in the core of the model ondary circulation in extremely intense Hurricane Gil- tropical cyclone (Fig. 9 of RE87), only small deviations bert (1988) by Dodge et al. (1999) shows an in¯ow from this congruency would be enough to produce a layer extending from the surface to 8 km near the eye- buoyant updraft. The presence of a small amount of wall. buoyancy in the steady-state nonhydrostatic tropical cy- A relatively deep in¯ow layer in the steady state is, clone would indicate that the model of a slantwise neu- however, predicted by the nonhydrostatic cloud model tral hurricane is not entirely correct. That the mainte- of Rotunno and Emanuel (1987, hereafter RE87). The nance of the steady-state vortex is dependent on the steady-state ®elds from this model indicate an accel- presence of convective instability in the eyewall has erating updraft in the lowest 10 km of the eyewall. This been hypothesized recently by W. M. Gray (1997, un- structure cannot be explained by neutrality to slantwise published manuscript). Gray argues that the intensity of convection. Since weak midlevel in¯ow is present in a tropical cyclone is limited by a balance between fric- the steady state, this suggests that entrainment is oc- tion and the maximum updraft generated by buoyancy

FIG. 7. Steady-state two-dimensional ®elds of (a) radial velocity (m s Ϫ1; contour interval, 5 m sϪ1) and (b) vertical velocity (m sϪ1; contour interval, 2 m sϪ1), from hydrostatic model of Emanuel (1995a).

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FIG. 9. Radial pro®les from Ooyama's (1969) numerical model: (a) tangential velocity (m s Ϫ1), (b) relative vorticity (sϪ1), (c) vertical velocity (m sϪ1), and (d) radial velocity (m sϪ1) without vorticity bump added at t ϭ 100 h.

and frictional convergence. However, since the steady- ever, does not suggest this is the case. The composite state results of the Emanuel's time-dependent models soundings from Frank (1977) show a relative humidity (1987, 1995b) agree closely with the MPI predicted of 95% at the innermost compositing radius (78 km), from the moist neutral theory and the results from RE87, decreasing to 90% at 222 km and 85% at 444 km. This one may argue that a large majority of the important suggests that the relative humidity in a tropical cyclone dynamics needed to determine the MPI are captured in is far from constant with radius, with boundary layer the simple theory. values in the environment typically above 80%. In fact, The assumption of constant boundary layer relative the composite surface relative humidity at a radius ex- humidity from the environment to the eyewall is also ceeding 1300 km is 82%, still above the value assumed of some concern. E86 cites the composite analysis of by Emanuel. Recent composite analysis of buoy and Frank (1977) to support the claim that the relative hu- Coastal-Marine Automated Network data (Cione et al. midity in the boundary layer is relatively constant with 2000) similarly shows that relative humidity is not con- radius outside of the eyewall of tropical cyclones and stant with radius. This study reveals relative humidities that its value is approximately 80%. Frank (1977), how- greater than 92% within 110 km of the center, with a

Unauthenticated | Downloaded 09/25/21 08:17 AM UTC JULY 2001 CAMP AND MONTGOMERY 1713 maximum just over 96% at 56 km, dropping to roughly gomery (1999) showed that convectively generated 84% outside of 200 km. It is clear then that the boundary vortex Rossby waves excited near or within the vortex layer approximation implemented by Emanuel is not RMW will act to spin up the vortex. If convectively ideal. generated vortex Rossby waves are excited too far from An illuminating situation develops, however, when the RMW, however, then the outer tangential winds the Emanuel boundary layer of constant surface tem- will be strengthened rather that the eyewall winds (see perature is compared with an observed hurricane bound- Table 3 of Montgomery and Enagonio 1998). The latter ary layer that is nonisothermal and the relative humidity process, when coupled to the boundary layer and con- increases inward. Consider Hurricane Mitch (1998). We vection, may explain the formation of secondary eye- can take the observed surface temperature of the en- walls (Montgomery and Kallenbach 1997). Although vironment from Mitch of 301 K and calculate the equiv- further observations and full-physics modeling studies alent potential temperature for the observed pressures are needed to obtain a complete understanding of the underneath the eyewall [ϳ969 mb from U.S. Air Force physics responsible for their formation, we shall nev- reconnaissance (Data obtained from National Hurricane ertheless suggest that the presence of outer vorticity Center's FTP site)] using an assumed value of relative bands or secondary eyewalls represent an important humidity of 80% (following Emanuel). The resulting intensity limiting mechanism. The formation of an out- value of ␪e is 364 K. Calculating the equivalent potential er vorticity band or secondary eyewall will be param- temperature directly from the observed surface temper- eterized here by impulsively adding a ®nite-amplitude ature, pressure, and relative humidity (298.75 K, 969 vorticity ring (or ``bump'') to the vorticity ®eld of an mb, 97%, respectively) from dropsondes deployed in axisymmetric tropical cyclone model during the de- the eyewall of Hurricane Mitch yields a value of ␪e velopment phase of the storm. averaging 364 K, the same as the ␪e calculated with the We have chosen Ooyama's (1969) tropical cyclone Emanuel boundary layer. This suggests that the ob- model to examine the possible effects of outer eyewall served boundary layer cooling and high relative hu- formation on maximum intensity. Ooyama's (1969) midity at the eyewall may tend to offset each other in model is a simple, axisymmetric, three-layer represen- terms of ␪e. Since it is the entropy de®cit (proportional tation of the hurricane vortex with parameterized con- to ␪e de®cit) that fuels the storm in Emanuel's model, vection. The control run of the model begins with a 10 the assumption of constant relative humidity and surface msϪ1 initial vortex possessing a 50-km RMW. The ini- temperature in the boundary layer allows the entropy tial vortex intensi®es, reaching hurricane intensity by structure predicted by Emanuel to be roughly similar to 70 h and a maximum intensity of 64 m sϪ1 at 130 h. that of observed tropical cyclones. In other words, the After reaching peak intensity, the vortex spins down Emanuel boundary layer seems to be robust to realistic slowly due to radial diffusion and a boundary layer variations in relative humidity and surface temperature, formulation that assumes the tangential wind is in gra- at least for current conditions. dient balance with the geopotential. During the evolution, the radius of maximum winds contracts to a radius of 40 km before expanding during the slow spindown 5. Internal limits to MPI: Convectively forced phase. No secondary wind maximum develops during vortex Rossby waves and secondary eyewalls the control run. Although a true steady state is never Since Emanuel uses the basic assumption of axi- attained, the model results prior to the weakening phase symmetry, the effects of asymmetric processes cannot are believed to be representative of what a more accurate be simulated explicitly but must be parameterized. hurricane model would predict. While the axisymmetry assumption captures many of Figure 9 shows the radial pro®le of tangential velocity the important processes that govern the evolution of (Fig. 9a), relative vorticity (Fig. 9b), vertical velocity tropical cyclones, it is nevertheless possible that asym- (Fig. 9c), and radial velocity (Fig. 9d) at t ϭ 100hof metric processes could, under certain conditions, limit the control run. At this time, the vortex is intensifying the intensity of the vortex relative to its axisymmetric rapidly. Each of the ®elds shown contains one distinc- MPI. Results from Project STORMFURY have sug- tive peak near the RMW. To simulate the impact of outer gested the important role played by concentric eyewalls vorticity bands or secondary eyewalls on intensi®cation, on hurricane intensity (Willoughby et al. 1985). The a Gaussian bump is added to the vorticity pro®le at t importance of the radial location of convection on hur- ϭ 100 h. The maximum vorticity of the bump is 10Ϫ3 ricane intensity has also been investigated [e.g., Ro- sϪ1 and has an e-folding radius from the center of the senthal and Moss (1971); see Anthes (1982) for a de- anomaly of 6 km. Such a bump would be representative tailed review up to 1980 and more recently by Bister of a symmetrical outer tangential wind maximum in a (2001), as well as Guinn and Schubert (1993) and tropical cyclone. The character of the bump is shown Schubert and Hack (1982) who interpret the negative in Fig. 10, which is identical to Fig. 9 except with the effect of outer core convection based on inertial sta- addition of the bump to the relative vorticity pro®le. In bility]. Montgomery and Kallenbach (1997), Mont- this particular experiment, the bump is added at a radius gomery and Enagonio (1998), and Moller and Mont- of 80 km. [Since the model is a balance model, the

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FIG. 10. As in Fig. 9 except with vorticity bump added at t ϭ 100 h. associated mass ®eld (geopotential) and meridional cir- leading to a spindown of the interior vortex. This in- culation is obtained directly upon inversion.] Adding terpretation is essentially the same as the STORMFURY the bump to the vorticity ®elds results in secondary hypothesis (see Anthes 1982, chapter 5) but intrinsic maxima in the tangential, vertical, and radial velocity convective/vorticity dynamics are being called upon ®elds. The model is then integrated forward in time. here for initiating the secondary eyewall. The impact of the bump on the model evolution is quite The amplitude and radial position of the vorticity dramatic, as summarized by Fig. 11 at t ϭ 110 h. While bump also plays an important role in the evolution of the secondary maxima in the tangential wind has been the vortex in Ooyama's model. Figure 12 shows the time suppressed, the intensi®cation at the original RMW has evolution of the layer 1 maximum tangential winds for been halted, and the maximum tangential winds have the control run as well as for several different radial actually decreased to below 40 m sϪ1. Also noteworthy placements of the vorticity bump. In each case, the in- is the fact that the secondary ``eyewall'' has become tensity of the vortex decreases immediately following dominant in the vertical velocity and radial velocity the insertion of the vorticity bump. The impact is found ®elds. In effect, the outer vorticity maximum appears to be more substantial as the radius of insertion moves to have robbed the inner wind maximum of its in¯ow, outward. When the bump is inserted at r ϭ 80 km or

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FIG. 11. As in Fig. 9 except at t ϭ 110 h (10 h after vorticity bump introduced). r ϭ 100 km, the maximum intensity of the vortex is 6. Conclusions reached at the time of insertion. For the case with the bump inserted at r ϭ 60 km, the vortex reaches max- Over the years there have been numerous efforts to imum intensity approximately 30 h later. These exper- establish a useful theoretical limit on the maximum in- iments suggest that the internal dynamics of convec- tensity of tropical cyclones. Such a limit would allow tively generated vortex Rossby waves, their coupling to for strides to be made in the dif®cult task of intensity the boundary layer, and formation of vorticity bands or forecasting. With a perfect theory for MPI, the observed secondary eyewalls may play a crucial role in limiting maximum intensities of tropical cyclones would reach the intensity of hurricanes, possibly providing the ul- the theoretical limit. That this is not the case, however, timate factor in prohibiting the formation of hypercanes. suggests that there are great strides yet to be made. Further analysis along similar lines with more compre- The theories of Miller (1958) and Holland (1997) hensive axisymmetric and three-dimensional hurricane appear to be the weakest of the MPI formulations. The models with and without ocean coupling is needed for heavy reliance on CAPE in the ambient atmosphere ap- a more complete understanding of this limiting mech- pears to be their main weakness. Modeling results using anism. Ooyama's (1969) model, as well as the models of Ro-

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(Frank 1977; Dodge et al. 1999), however, suggest that moderate in¯ow occurs through a relatively deep layer extending well above the boundary layer and that ``near boundary layer'' in¯ow, alone, may not be suf®cient to maintain a tropical cyclone against friction. The deep in¯ow observed and modeled in a cloud resolving model indicates that the assumption of neutrality to slantwise convection breaks down to some extent in the steady state. But considering that the predicted MPI correlates well with the RE87 nonhydrostatic results, it would appear that the breakdown of the slantwise neutrality assumption is not entirely detrimental. The WISHE model developed by Emanuel (1986±97) FIG. 12. Evolution of maximum tangential winds in Ooyama's contains a unique instability that produces extraordi- (1969) model with no vorticity bump (solid), vorticity bump at r ϭ 60 km (dash±dot), vorticity bump at r ϭ 80 km (dash), and narily intense tropical cyclones with tangential winds vorticity bump at r ϭ 100 km (dot). approaching 300 m sϪ1, corresponding to a Mach number of 0.86. This occurs when the rise in the surface mixing ratio as the pressure lowers under the eyewall tunno and Emanuel (1987), and Emanuel (1989, 1995a) is more than is needed to maintain the eyewall surface suggest that ambient convective instability plays a minor pressure, leading to runaway intensi®cation. Such hy- role in the ®nal intensity of the tropical cyclone. Thus, percanes are, in fact, realized in the axisymmetric cloud the physical basis for determining MPI used by Holland model of RE87 starting from an ambient atmosphere (1997) and Miller (1958) seems not well founded. Ad- that is neutral to slantwise convection. The Holland ditionally, each of the latter formulations relies on the model, however, does not appear to produce the same parameterization of the thermodynamic pro®le in the type of runaway vortex for a convectively neutral en- eye. Observations of the thermodynamic pro®les vironment. Extremely low surface pressures in this mod- through the full depth of tropical cyclone eyes are rare, el seem to be realized only with very high surface tem- so the actual thermodynamic structure is not yet well peratures in a high-CAPE environment. This difference known. With such little knowledge of the structure of between the two theories indicates a fundamental dif- the tropical cyclone eye, any attempt to parameterize ference in the underlying physics that maintains tropical the thermodynamics that would correspond to the stron- cyclones. It is unclear, however, whether or not such gest tropical cyclones is quite arbitrary. The combina- hypercanes predicted by Emanuel's model can materi- tion of the erroneous dependence upon ambient con- alize under more realistic three-dimensional scenarios. vective instability and the arbitrary eye sounding leads Using Ooyama's (1969) tropical cyclone model, we to the unrealistic predictions of MPI from Holland have shown that vorticity bands or secondary eyewalls (1997) for sea surfaces temperatures greater than 302 K. excited in an intensifying tropical cyclone may provide Modeling results showing the minor role of ambient a powerful constraint on the actual vortex intensity and convective instability in the tropical cyclone develop- may ultimately prohibit the formation of hypercanes, ment indicate that MPI formulations that do not rely on despite favorable environmental conditions. While these such instability may be closer to reaching the desired results do not include the effects of vertical wind shear result. The convectively neutral formulation proposed explicitly, the effect of wind shear is often to excite by Emanuel (1986) appears to generate a useful upper convection away from the center. Thus, we believe these bound on tropical cyclone intensity for the range of sea results are insightful even though they are carried out surface temperatures examined. Additionally, although in an axisymmetric framework. Further analysis of this Emanuel's boundary layer parameterization in the outer issue awaits future work. regions of tropical cyclones is highly simpli®ed, the Despite the best attempts to formulate a limit on trop- predicted entropy structure of the boundary layer is ical cyclone intensity, current efforts still fall short of roughly similar to the observed boundary layer entropy providing an effective prediction of MPI that can be structure of observed tropical cyclones. used with con®dence in an operational setting. While There are, however, some basic aspects of the tropical the WISHE theory of Emanuel appears to predict a cyclone structure predicted by Emanuel's theories that storm structure that is close to reality (Emanuel 1999), are somewhat suspect. All of the in¯ow in the theoretical there are potentially signi®cant shortcomings that need model is assumed to occur within or just above the to be examined more critically. There are numerous pa- boundary layer. This implies that the angular momentum rameters that have yet to be accurately accounted for in imported in this relatively thin layer is suf®cient to com- the reviewed theories that should increase their useful- bat the detrimental effects of friction and maintain a ness. A better account of the role of asymmetric vortex steady state. Nonhydrostatic cloud model simulations dynamics in the intensi®cation and maintenance of trop- (RE87) and observations of intense tropical cyclones ical cyclones is needed. The use of upper-ocean heat

Unauthenticated | Downloaded 09/25/21 08:17 AM UTC JULY 2001 CAMP AND MONTGOMERY 1717 content and a better understanding of the role of ocean coef®cients and a revised steady-state model incorporating eye cooling may aid in producing an MPI formulation that dynamics. J. Atmos. Sci., 52, 3969±3976. ÐÐ, 1997: Some aspects of hurricane inner-core dynamics and en- is more representative of the intensities which storms ergetics. J. Atmos. Sci., 54, 1014±1026. actually achieve (Shay et al. 1992). In addition, obser- ÐÐ, 1999: Thermodynamic control of hurricane intensity. Nature, vations of the actual boundary layer structure of tropical 401, 665±669. cyclones through the use of Global Positioning System ÐÐ, K. Speer, R. Rotunno, R. Srivastava, and M. Molina, 1995: Hypercanes: A possible link in global extinction scenarios. J. dropsondes promise to add important information about Geophys. Res., 100, 13 755±13 765. the interaction between the ocean and the storm cir- Frank, W. M., 1977: The structure and energetics of the tropical culation and help quantify how much heat exchange is cyclone I. Storm structure. Mon. Wea. Rev., 105, 1119±1135. realized between the underlying ocean and atmosphere. Gray, S., 1994: Theory of mature tropical cyclones: A comparison between Kleinschmidt (1951) and Emanuel (1986). JCMM Rep. 40, 50 pp. [Available from Joint Centre for Mesoscale Meteo- Acknowledgments. This research was funded in part rology, University of Reading, P.O. Box 240, Reading, Berkshire by NSF Grant 9732678 and Colorado State University. RG6 2FN, United Kingdom.] Guinn, T. A., and W. H. Schubert, 1993: Hurricane spiral bands. J. Special thanks to go Kerry Emanuel for providing easily Atmos. Sci., 50, 3380±3403. accessible source code for his MPI theories and nu- Hawkins, H. F., and D. T. Rubsam, 1968: Hurricane Hilda, 1964. 2. merical model, as well as Weiqing Qu and Greg Holland Structure and budgets of the hurricane on October 1, 1964. 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