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Home , 1932 Bahamas hurricane, Hurricane Alicia, Hurricane Andrew, Hurricane Hugo, Mesovortices, Typhoon Ida (1945)

in : Dynamics of a Disaster ^

H. E. Willoughby and P. G. Black Hurricane Research Division, AOML/NOAA, , Florida

ABSTRACT

Four meteorological factors aggravated the devastation when Hurricane Andrew struck : completed replacement of the original eyewall by an outer, concentric eyewall while Andrew was still at sea; translation so fast that the crossed the populated coastline before the influence of land could weaken it appreciably; extreme speed, 82 m s_1 measured by aircraft flying at 2.5 km; and formation of an intense, but nontornadic, convective vortex in the eyewall at the time of . Although Andrew weakened for 12 h during the eyewall replacement, it contained vigorous and was reintensifying rapidly as it passed onshore. The just offshore was warm enough to support a pressure 20-30 hPa lower than the 922 hPa attained, but Andrew hit land before it could reach this potential. The difficult-to-predict mesoscale and vortex-scale phenomena determined the course of events on that windy morning, not a long-term trend toward worse hurricanes.

1. Introduction might have been a harbinger of more devastating hur- ricanes on a warmer globe (e.g., Fisher 1994). Here When Hurricane Andrew smashed into South we interpret Andrew's progress to show that the ori- Florida on 24 August 1992, it was the third most in- gins of the disaster were too complicated to be ex- tense hurricane to cross the coastline in plained by thermodynamics alone. the 125-year quantitative climatology. It destroyed $25 billion in property and killed 15 people directly (Mayfield et al. 1994; Rappaport 1994). Andrew's 2. Hurricane intensification landfall coincided with a period of historically high global temperatures (Follard et al. 1992) and followed Hurricanes, by definition, sustain surface winds two other memorable hurricanes: Gilbert in 1988, greater than 33 m s_1. Their intensity is measured in which took 318 lives and established 888 hPa as a terms of maximum wind or minimum sea level pres- record for the lowest hurricane sea level pressure in sure at the storm center/^. The Saffir-Simpson scale the Atlantic basin (Lawrence and Gross 1989); and (Simpson 1974) ranks increasing intensity in catego- Hugo in 1989, which took 49 lives and destroyed ries 1-5. The threshold of category 5 is sustained (1- $7 billion in property, the largest total up to that time min average) surface wind stronger than 69 m s_1 or (Case and Mayfield 1990). These events, reinforced Pc below 920 hPa. At inland sites, the surface winds by thermodynamic arguments that relate the hurri- used to assign categories are typically 65% of the wind canes' maximum intensity to measured at several kilometers altitude by aircraft. (Emanuel 1987), raised public concern that Andrew The weaker wind inland in the frictional boundary layer is caused by the greater aerodynamic roughness of the land surface. The reduction occurs over the first Corresponding author address: Dr. H. E. Willoughby, Hurricane Research Division, AOML/NOAA, 4301 Rickenbacker Cause- few kilometers inland. At sea or on the coast, the sur- way, Miami, FL 33149. face winds are stronger, 70%-80% of the flight-level E-mail: [email protected] wind, but the strongest gusts may approach the flight- In final form 15 September 1995. level wind (Powell 1982, 1987; Powell et al. 1991;

Bulletin of the American Meteorological Society 543

Unauthenticated | Downloaded 10/08/21 06:01 AM UTC Black 1993). Thus, the sustained surface wind equiva- fall, the smaller effective heat capacity of the solid lent to the 82 m s_1 flight-level wind observed in An- surface leads to boundary layer cooling, reduced drew would be 53 m s_1 ashore and 57-66 m s-1 at sea. evaporation, and weakening of the storm as a whole Independent estimates of Andrew's strongest winds (Miller 1964; Tuleya 1994). This thermodynamically agree generally with these figures (Rappaport 1994), induced spin-down of the vortex is more gradual than but debris patterns at some inland sites are consis- the frictional reduction of the surface wind at the coast. tent with surface winds near the flight-level value A hurricane that has achieved the greatest intensity (Wakimoto and Black 1994). possible at sea may be modeled as a heat engine oper- Category 5 hurricanes are rare in the Atlantic. The ating between a warm reservoir at the sea surface tem- historical record shows U.S. landfall by only two, perature TS and a cold reservoir at the tropopause Camille in 1969 and the Labor Day Hurricane in 1935. temperature Tr Figure 1 shows that Ps, the lowest pos- Category 4 hurricanes cross the U.S. coastline more sible/^, becomes lower as the thermodynamic efficiency often, every 6-7 years on average. No hurricane with (TS - TT)/TS increases (Emanuel 1988). An independent Pc < 900 hPa has reached the U.S. mainland since empirical estimate of Ps can be made by tabulating the 1 1900, although three have occurred in the Atlantic greatest intensity ever observed as a function of Ts basin (Hebert et al. 1992). Northwestern-Pacific ty- (DeMaria and Kaplan 1994). An example of this es- phoons reach category 4 and 5 intensities more fre- timate also appears in Fig. 1. The thermodynamic es- quently. Rapid deepening, defined as a rate of pressure timate of Ps falls 10 hPa for a 1°C increase in Ts near fall > 42 hPa in 24 h, is the mode of intensifica- 30°C, but the empirical estimate falls 17 hPa per de- tion in three quarters of the that reach Pc gree. The more rapid fall of empirical Ps stems from < 920 hPa and in all that reach Pc < 900 hPa (Holliday a climatological correlation between TS and TT\ the and Thompson 1979). The most intense recent Atlan- tropopause is higher and colder in low latitudes where tic hurricanes (Camille, Allen, Gloria, Gilbert, and the sea is warmer. Figure 1 conveys an illusory impres- Hugo) reached their lowest surface pressures after a sion of precision. Sea surface temperature is typically day or two of rapid deepening. Rapid deepening of- known to ±1°C, and its range in the Tropics is 4-5°C. ten follows interactions between tropical and The high-intensity ends of the curves are supported midlatitude troughs or subtropical lows. The mecha- by only a handful of cases. Indeed, one interpretation nism involves secondary circulations forced by upper- of the observations is that Ts is not the controlling fac- tropospheric convergence of angular momentum tor in intensity of most hurricanes (Evans 1993). On (Pfeffer 1958; Pfeffer and Challa 1981,1992; Holland any given day during hurricane season, much of the and Merrill 1984; Molinari and Vollaro 1989; DeMaria et al. 1993). The dynamics of the eddies are not well understood, but it is clear that the momen- tum convergence forces axisymmetric just below the tropopause with deep compensating inflow in the lower troposphere joined to the outflow by broad ascent around the hurricane's center. The as- cent causes intensification by destabilizing the column and enhancing convective heat release. Hurricanes' low pressures are driven by thermody- namic disequilibrium between the ocean and the at- mosphere. Evaporation from the sea supplies latent heat to the surface boundary layer that, in turn, feeds cumulonimbus convection around the eye. Entrain- ment into the convection forces a midlevel flow that concentrates angular momentum and spins up the FIG. 1. Minimum sea level pressure attainable in hurricanes vortex (Ooyama 1969, 1982). After hurricane land- as a function of sea surface temperature for tropopause temperatures -65°, -70°, and -75°C based upon Emanuel's (1988) argument (dotted curves) and derived form DeMaria and Kaplan's (1994) observational study (solid curve). Actual 1 The Labor Day Storm, 892 hPa; Allen of 1980, 899 hPa; and minimum sea level pressures at greatest intensity for Hurricanes Gilbert. Gilbert, Hugo, and Andrew are designated G, H, and A.

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Unauthenticated | Downloaded 10/08/21 06:01 AM UTC tropical Atlantic is warm enough to produce a cat- Ps for the prevailing Ts, 30°C. Over at egory 4 or 5 hurricane, but few occur. 2045 UTC on 23 August, a reconnaissance aircraft The most important factor that limits both inten- reported a second eyewall at 20-km radius concen- sity of individual hurricanes and total activity on a sea- tric with the original eye, a double flight-level wind sonal basis is vertical shear of the environmental wind maximum (Fig. 3c), and a rise of Pc to 927 hPa. The (Gray 1968, 1984). Shear raises Pc by "ventilating" Miami radar, operating at extreme range, confirmed the warm vortex core that supports lowered hydro- the concentric eyewalls. By 0410 UTC on 24 August, static surface pressure (Simpson and Riehl 1958). when only the outer eyewall remained (Fig. 3d), Pc Another factor that limits intensity in low-shear situa- had risen to 941 hPa. Andrew then tracked due west tions is weakening due to replacement of a preexis- toward landfall over Homestead, Florida; the new eye tent eyewall by a newly formed outer concentric contracted to 13 km; and Pc fell to 936 hPa. Sea level eyewall. Both inner and outer eyewalls are contract- pressure at the storm center continued to fall, reach- ing rings of deep cumulonimbus that coincide with ing 922 hPa again after landfall. The aircraft encoun- local maxima of the swirling wind. Convergence in tered graupel, severe turbulence, and spectacular the frictional boundary layer localizes convection near electrical displays—both meteorological and from the the wind maximum. The convective updrafts entrain failing municipal power grid—as it measured a reduc- air outward from the area encompassed by the ring, tion of maximum flight-level wind from 82 m s-1 one leading to subsidence, adiabatic warming, and pres- hour before landfall (Fig. 3e) to 77 m s_1 at landfall sure falls (Shapiro and Willoughby 1982). The tight- (Fig. 3f) and to 70 m s_1 one hour after landfall (Fig. ening gradient of pressure across the eyewall increases 3g). Nevertheless, the strongest surface gusts re- the wind at the maximum and just inward from it, so mained close to 80 m s"1, consistent with the continu- that the maximum strengthens and propagates toward ing pressure fall (Wakimoto and Black 1994). the vortex center, carrying the convection with it. Earlier, after the original eyewall's replacement and Hurricanes with winds stronger than 50 m s"1 often dissipation, the Miami radar had recorded a pair of in- weaken abruptly when an outer concentric eyewall re- tense echoes orbiting around the center on opposite places the inner.2 Typically, at least 24 h elapse before sides of the new eyewall. As the storm approached, such can recover their former intensity, and many cyclonic appeared in the eyewall adjacent never do (Willoughby et al. 1982; Willoughby 1990). to the reflectivity maxima. When the eyewall reached the coast, the vortex pair stopped rotating and intense convection erupted in the onshore winds north of the 3. Andrew center, causing a fixed cross-track asymmetry to re- place the rotating vortex pair. Before landfall, the air- Andrew's precursor was a that craft data show a 21 m s_1 asymmetry between the wind formed over Africa and traversed the tropical Atlan- tic. It became Tropical Storm Andrew (wind stron- ger than 17 m s_1) on 17 August 1992 but showed little sign of intensification until 21 August, when it be- gan to deepen rapidly as it moved into a region of low shear and warm water. The rapid deepening began af- ter Andrew encountered an upper-level subtropical low, but the chain of cause and effect is unclear (Mayfield et al. 1994; Rappaport 1994). In less than 48 h, Pc fell from 1014 to 922 hPa as the radius of the original eye contracted from 22 to 8 km (Fig. 2). Dur- ing this time, the wind distribution changed from a flat profile with a 22 m s_1 maximum (Fig. 3a) to a sharply peaked profile with a maximum of nearly 1 70 m s" (Fig. 3b). The lowest Pc was - 25 hPa above FIG. 2. Andrew's minimum sea level pressure (solid curve), minimum attainable (DeMaria and Kaplan 1994) sea level pressure (dashed curve), and radii of the original (I) and outer (II) • For example, Gloria of 1985 and Allen. eyewalls as functions of time (dotted curves).

Bulletin of the American Meteorological Society 545

Unauthenticated | Downloaded 10/08/21 06:01 AM UTC in the northern eyewall, where Andrew's swirling wind moved over southern Dade County, halfway around and translation combined, and the southern eyewall, the eye to south of Homestead, where they dissipated where they were opposed. This asymmetry, which was in the southern eyewall (Fig. 4). Each had a life cycle already greater than could be explained by twice the of 8-15 min during which the radar-derived rain rate 8.5 m s_1 translation, increased to 26 m s_1 over land. increased from 10 to 150 mm h_1. Two generally ac- Even as the wind destroyed the Miami radar, An- cepted explanations for increased surface winds in drew came within range of radars at Tampa and heavy convective rain are that precipitation-induced Melbourne, Florida (NOAA 1993). Because the radars downdrafts inject high-velocity air from the free at- were ~ 300 km from the hurricane, their beams inter- mosphere into the frictional boundary layer and that sected the eyewall at 3-8-km altitude. At intervals of the surface wind accelerates even more as the "down- 3-7 min during the 45 min required for the eye to move burst" (Caracena and Maier 1987) spreads along the ashore, seven intense convective cells formed where ground. In Andrew, where the most severe damage lay the northern eyewall intersected the coast. The cells in streaks along and downwind of the convective cells'

FIG. 3. East to west profiles of observed by aircraft in Hurricane Andrew (a) early in the initial deepening (21 August at 1650 UTC); (b) at the end of the first episode of deepening, before the eyewall replacement (23 August, 1655 UTC); (c) during the eyewall replacement (23 August, 2045 UTC); and (d) after the eyewall replacement (24 August, 0220 UTC). The original and new eyewalls are indicated by Roman numerals. South to north profiles of wind (e) an hour before landfall (24 August, 0810 UTC); (f) at landfall (0915); and (g) an hour after landfall (1020).

546 Vol. 77, No. 3, March 1996 Unauthenticated | Downloaded 10/08/21 06:01 AM UTC trajectories around the eye, the seem to have caused 20 m s_1 surges in wind speed on a > 60 m s-1 basic flow (Wakimoto and Black 1994; Fujita 1993). A third, not necessarily exclusive, explanation for the damage streaks lies in the dynamics of the updrafts. The convective locus of cell formation became fixed where the north eyewall intersected the coast because increased friction enhanced surface convergence there. The cells' rapid growth implied strong updrafts that locally increased the large back- ground vorticity through stretching of its vertical com- ponent and tilting of its horizontal component into the vertical. This process appears to have produced at least one 3-5-km-diameter mesovortex, similar to those ob- served in some severe (Davies-Jones 1986). The polar-orbitingNOAA-11 satellite observed a "hot spot" that apparently formed as the convective updrafts in the northern eyewall drew air from the eye near the surface, producing descent, adiabatic warm- ing, and a local hydrostatic pressure minimum. Poststorm calibration of surface pressure observations taken by resolute amateur meteorologists (Rappaport 1994) showed, with remarkable consistency, that the lowest pressure occurred at the inner edge of the eye- wall within 3 min of the satellite image and coincided with its warmest pixel. This pressure was 9 hPa below the simultaneous minimum pressure extrapolated at the center of the eye from the reconnaissance aircraft. Mesovortices have been described in other eyewalls. High-altitude photographs of Ida from U2 aircraft in 1958 show the eyewall clouds folding into the eye as they wrap around a mesovortex (Fletcher et al. 1961). In , a research aircraft was nearly lost when it penetrated a meso- vortex. Flight-level measurements showed a 30 m s_1 relative circulation around 13 hPa local pressure mini- mum moving with the 70 m s-1 eyewall flow. The vortex orbited the eyewall five times in 1.5 h as a suc- cession of convective cells nearly identical with those in Andrew developed and decayed (Black and Marks 1991). At the times these mesovortices formed, Ida and Hugo, like Andrew, were already intense and continuing to intensify rapidly. FIG. 4. Individual sweeps from the Tampa WSR-57 radar showing a convective cell forming in Andrew's northern eyewall 4. Conclusions and moving around the eye across South Florida. F2 damage (failure of windows and doors, partial collapse of roofs) was common south of 24°40'N in the path of the eyewall (Fujita 1993). Boxes indicate Preseason forecasts called for below-average hur- sites of F3 damage (roofs completely blown off, walls collapse). ricane activity during 1992. Measured in terms of sta- The circle indicates the position of the minimum observed surface tistics such as the number of hurricanes or the number pressure. Time (UTC) appears in the lower-right corner of each of days on which hurricanes occurred, the forecasts frame.

Bulletin of the American Meteorological Society 547

Unauthenticated | Downloaded 10/08/21 06:01 AM UTC proved correct (Gray 1992). Andrew's landfall was the , and F. D. Marks, 1991: The structure of an eyewall meso- exception in an otherwise inactive season. vortex in hurricane Hugo (1989). Preprints, 19th Conf. on Hurricanes and Tropical , Miami, FL, Amer. The event itself embodied a complex interplay of Meteor. Soc., 579-582. competing effects. Low shear and warm water east Caracena, F., and M. W. Maier, 1987: Analysis of a microburst of Florida were necessary, but by no means sufficient, in the FACE meteorological mesonetwork in South Florida. conditions for this disaster. Although the ocean was Mon. Wea. Rev., 115, 969-985. unusually warm, Gilbert and Hugo (Fig. 1) were over Case, R., and M. Mayfield, 1990: season of 1989. Mon. Wea. Rev., 118, 1165-1177. cooler water—albeit in lower latitudes where the trop- Davies-Jones, R. P., 1986: dynamics. opause was higher and colder—when they reached Morphology and Dynamics, E. Kessler, Ed., University of Pc below Andrew's. Probably the most important Press, 197-236. single factor in Andrew's intensification was the DeMaria, M., and J. Kaplan, 1994: Sea surface temperature and poorly understood process that started the rapid deep- the maximum intensity of Atlantic tropical cyclones. J. Cli- ening. Extrapolation of the deepening trend past the mate, 7,1324-1334. , J.-J. Baik, and J. Kaplan, 1993: Upper-level eddy angular eyewall replacement to landfall yields Pc ~ 900 hPa, momentum fluxes and tropical- intensity change. close to Ps. Without the filling during the replacement, J. Atmos. Set, 50, 1133-1147. Andrew might have become stronger than Camille Emanuel, K. A., 1987: The dependence of hurricane intensity on and rivaled the Labor Day Hurricane of 1935. On the climate. Nature, 326, 483-485. other hand, if the replacement had followed a course , 1988: The maximum intensity of hurricanes. J. Atmos. Sci., typical of other cases, Andrew might have weakened 45,1143-1155. Evans, J. L., 1993: Sensitivity of tropical cyclones to sea surface to category 3 by landfall and the mesovortex would temperature. J. Climate, 6, 1133-1140. never have formed. If the storm had moved more Fisher, D. E., 1994: The Scariest Place on Earth. Random House, slowly, it would have had more time to reintensify 250 pp. over the Gulf Stream. But the translation's contribu- Fletcher, R. D., J. R. Smith, and R. C. Bundgaard, 1961: Supe- tion to the most destructive winds in the north eyewall rior photographic reconnaissance of tropical cyclones. Weatherwise, 14, 102-109. would have been less, and more time would have been Follard, C. K., T. R. Karl, N. Nicholls, B. S. Nyenzi, D. E. Parker, available for spin-down over land (Figs. 3e-g). The and K. Ya. Vinnikov, 1992: Observed climate variabil- worst damage might then have been confined to a ity and change. Climate Change 1992, J. T. Houghton, narrow strip along the coast. The fast motion and in- B. A. Callander, and S. K. Varney, Eds., Cambridge Univer- tense convection appear to have been essential to the sity Press, 135-170. eyewall mesovortex that caused the worst devastation. Fujita, T. T., 1993: Damage survey of Hurricane Andrew in South Florida. NOAA , 34, 8, 25-29. Thus, we argue that reasons for the most expensive Gray, W. M., 1968: Global view of the origin of tropical distur- hurricane landfall in U.S. history lie in Andrew's con- bances and storms. Mon. Wea. Rev., 96, 669-700. vective and vortex-scale evolution rather than in a cli- , 1984: Atlantic seasonal hurricane frequency. Part I: El Nino matic trend toward more intense hurricanes. and the 30 mb quasi-biennial oscillation influences. Mon. Wea. Rev., 112, 1649-1668. Acknowledgments. We are grateful to P. P. Dodge and , 1992: Summary of 1992 Atlantic cyclone activity and M. L. Black who recorded the Miami and Tampa radars, to verification of the author's forecast. Issued 24 November. R. G. Carter for his account of the reconnaissance flight at land- [Available from Dept. of Atmos. Sci., Colorado State Univ., fall, to everyone at the U.S. Air Force Reserve 53 Weather Re- Fort Collins, CO 80523.] connaissance Squadron (then the 815th WRS) for flight-level Hebert, P. J., J. D. Jarrell, and M. Mayfield, 1992: The deadli- observations and for the skill and determination with which they est, costliest, and most intense hurricanes of this century. flew in Andrew, to W. M. Barry, M. L. Black, and M. E. Rahn NOAA Tech. Memo. NWS NHC-31,40 pp. [Available from for help with data and graphics, and to S. B. Goldenberg, National Hurricane Center, 1161 SW 17th St., Miami, FL J. M. Orient, M. D. Powell, L. J. Shapiro, and L. J. Willoughby 33165-2149.] for thoughtful comments on an earlier draft. HEW's contribution Holland, G. J., and R. T. Merrill, 1984: On the dynamics of tropi- was supported by ONR Grant N00014-94-F-0045. cal cyclone structure changes. Quart. J. Roy. Meteor. Soc., 110, 723-745. Holliday, C. R., and A. H. Thompson, 1979: Climatological char- References acteristics of rapidly intensifying typhoons. Mon. Wea. Rev., 107, 1022-1034. Black, P. G., 1993: Evolution of maximum wind estimates in ty- Lawrence, M. B., and J. M. Gross, 1989: Atlantic hurricane sea- phoons. Disasters, J. Lighthill, Z. Zhemin, son of 1988. Mon. Wea. Rev., 117, 2248-2259. G. Holland, and K. Emanuel, Eds., Peking University Press, Mayfield, M., L. Avila, and E. N. Rappaport, 1994: Atlantic hur- 104-115. ricane season of 1992. Mon. Wea. Rev., 122, 517-538.

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Unauthenticated | Downloaded 10/08/21 06:01 AM UTC Miller, B. I., 1964: A study of the filling of , P. P. Dodge, and M. L. Black, 1991: The landfall of Hurri- (1960) over land. Mon. Wea. Rev., 92, 389-406. cane Hugo in the : Surface wind distribution. Wea. Molinari, J., and D. Vollaro, 1989: External influences on hurri- Forecasting, 6, 379-399. cane intensity. Part I: Outflow layer eddy angular momentum Rappaport, E. N., 1994: Hurricane Andrew. Weather, 49,51-61. fluxes. J. Atmos. Sci., 46, 1093-1105. Shapiro, L. J., and H. E. Willoughby, 1982: The response of bal- NO A A, 1993: Natural Disaster Survey Report, Hurricane Andrew: anced hurricanes to local sources of heat and momentum. South Florida and , August 23-26, 1992. 131 pp. J. Atmos. Sci., 39, 378-394. [Available from National Oceanic and Atmospheric Adminis- Simpson, R. H., 1974: The hurricane disaster-potential scale. tration, , Silver Spring, MD 20910.] Weatherwise, 27, 169. Ooyama, K. V., 1969: Numerical simulation of the life cycle of , and H. Riehl, 1958: Mid-tropospheric ventilation as a con- tropical cyclones. J. Atmos. Sci., 26, 3-40. straint on hurricane development and maintenance. Proc., , 1982: Conceptual evolution of the theory and modeling of Technical Conf. on Hurricanes. Miami, FL, Amer. Meteor. the tropical cyclone. J. Meteor. Soc. , 60, 369-380. Soc., D4.l-D4.10. Pfeffer, R. L., 1958: Concerning the mechanics of hurricanes. Tuleya, R. E., 1994: Tropical storm development and decay: Sen- J. Meteor., 15, 113-120. sitivity to surface boundary conditions. Mon. Wea. Rev., 122, , and M. Challa, 1981: A numerical study of the role of eddy 291-304. fluxes of momentum in the development of Atlantic hurri- Wakimoto, R. M., and P. G. Black, 1994: Damage survey of canes. J. Atmos. Sci., 38, 2393-2398. Hurricane Andrew and its relationship to the eyewall. Bull. , and , 1992: The role of environmental asymmetries Amer. Meteor. Soc., 75, 189-200. in Atlantic hurricane formation. J. Atmos. Sci., 49,1051-1059. Willoughby, H. E., 1990: Temporal changes in the primary cir- Powell, M. D., 1982: The transition of the culation in tropical cyclones. J. Atmos. Sci., 47, 242-264. boundary-layer wind field from the open to , J. A. Clos, and M. G. Shoreibah, 1982: Concentric eye landfall. Mon. Wea. Rev., 110, 1912-1932. walls, secondary wind maxima, and the evolution of the hur- , 1987: Changes in the low-level kinematic and thermody- ricane vortex. J. Atmos. Sci., 39, 395-411. namic structure of (1983) at landfall. Mon. Wea. Rev., 115, 75-99.

^llll BACK TO THE FUTURE M+ Old and New Forecasting Techniques Workshop, 19 June 1996 Boston,

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550 Vol. 77, No. 3, March 7 996

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