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OCTOBER 2008 LAJOIEANDWALSH 1007

A Technique to Determine the Radius of Maximum of a Tropical

FRANCE LAJOIE AND KEVIN WALSH School of Sciences, University of Melbourne, Parkville, Victoria,

(Manuscript received 2 October 2007, in final form 22 January 2008)

ABSTRACT

A simple technique is developed that enables the of a to be estimated from data. It is based on the characteristic cloud and wind structure of the eyewall of a tropical cyclone, after the method developed by Jorgensen more than two decades ago. The radius of maximum wind is shown to be partly dependent on the radius of the and partly on the distance from the center to the top of the most developed cumulonimbus nearest to the cyclone center. The technique proposed here involves the analysis of high-resolution IR and microwave satellite imagery to determine these two parameters. To test the technique, the derived radius of maximum wind was compared with high-resolution wind analyses compiled by the U.S. National Hurricane Center and the Atlantic Oceano- graphic and Meteorological Laboratory. The mean difference between the calculated radius of maximum wind and that determined from observations is 2.8 km. Of the 45 cases considered, the difference in 50% of the cases was Յ2 km, for 33% it was between 3 and 4 km, and for 17% it was Ն5 km, with only two large differences of 8.7 and 10 km.

1. Introduction Another sensor on board a polar-orbiting satellite that can produce high-resolution surface wind fields r To determine m, the radius of maximum wind for a over the is the Wind Synthetic Aperture tropical cyclone, one needs to analyze the strong sur- Radar (WiSAR). It measures the small-scale ocean sur- face in its inner core and in its eyewall. There are face roughness from which can be determined the di- a number of methods that have been developed to rection and the speed of the surface wind (Lehner et al. measure or estimate surface winds from satellite sen- 2006). WiSAR waves can penetrate cloud layers and sors but they are not reliable in the inner core of a and can operate day and night in all con- tropical cyclone. The Special Sensor Microwave Imager ditions and are therefore well suited to determining the (SSM/I) can only be used in cloudless and cannot radius of maximum wind in a tropical cyclone. The only therefore be used within the problem with WiSAR is that the data are only available (CDO) or in the eyewall of a tropical cyclone (Good- when the tropical cyclone is along the satellite track and berlet et al. 1989). Satellite-based microwave scatter- are therefore available only twice a day. ometers can only produce a reasonably good estimate Surface winds inside the CDO are also obtainable of to moderate surface winds in areas of no or from reconnaissance planes that fly at an of slight (Weissman et al. 2002; Yueh et al. about 3 km. The measured 3-km winds are reduced to 2003). They are therefore not reliable for estimating the the surface by using an empirically derived relationship strong winds associated with tropical . Surface (Franklin et al. 2003). Using these flight data, the U.S. winds can also be estimated from satellite cloud track National Hurricane Center can determine the maxi- winds deduced from geostationary (Dunion mum surface wind and the radius of maximum wind. and Velden 2002; Velden et al. 2005), but these also are But these flight wind data are now only available in the only useful outside the CDO. North Atlantic when a hurricane is within the flight range of the reconnaissance aircraft. In the central North Atlantic and in other Corresponding author address: Dr. Kevin Walsh, School of Earth Sciences, University of Melbourne, Parkville, VIC 3010, there is no method to directly measure the maximum Australia. winds or the radius of maximum wind. E-mail: [email protected] Recently, Kossin et al. (2007) used reconnaissance

DOI: 10.1175/2008WAF2007077.1

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flight data to investigate the relationship that exists be-

tween the eye size (re) and rm. They assumed re to be the mean radius of the Ϫ45°C isotherm in the cloud-top brightness temperature analysis. Their mean absolute

error in determining rm was 4.7 km. Hsu and Babin (2005) have suggested that the radius of maximum winds in a tropical cyclone is the distance between the coldest cloud-top temperature surround- ing the eye and the warmest temperature in the eye. They evaluated their hypothesis on only one cyclone, however. Here, we describe below a simple and easy-

to-use technique for estimating rm from the color- enhanced imagery of high-resolution IR satellite cloud data, from microwave Tropical Rainfall Measuring Mission (TRMM) Microwave Imager (TMI; Simpson et al. 1988; Lonfat et al. 2004) and TRMM Precipitation Radar (PR) data (Iguchi et al. 2000).

2. A technique for determining rm from satellite cloud data Jorgensen (1984a,b) has studied the eyewall structure of tropical cyclones, using a comprehensive set of sat- ellite cloud pictures, photographs of eyewall , and radar and reconnaissance flight data along 146 flight legs across the eye and eyewall of five hurricanes. His schematic representation of a cross section of the eyewall of North Allen on 5 August 1980 is shown in Fig. 1a. More recent studies (e.g., Marks and Houze 1987; Marks et al. 1992; Corbosiero et al. 2005) have found rather similar structures in their analyses, although unlike in Fig. 1a downdrafts are also typically observed in regions close to the main updraft. In Fig. 1a a few letters have been added to help in the following discussion. We use “A” as the inner edge of the eyewall at the surface and “T” as the top of the most developed cumulonimbus nearest to the cyclone center. Let the coordinates of “A” and “T” be denoted,

respectively, by (re, 0) and (rt, ht) (see also Fig. 1b). The line AT is inclined to the horizontal by an angle ␸ that may be as large as 45°–75° for intense with small

eye radius (re) and rm, and as small as 25° for weaker storms or for storms with large re and rm (Jorgensen 1984a). Let the straight line AT be inclined to the hori- ␸ zontal by an angle . The maximum updraft wm within this cumulonimbus is along BC, which represents the FIG. 1. (a) A schematic cross section of eyewall features, after Jorgensen (1984b). The schematic depicts the locations of the variation of wm with height in the low and middle levels, clouds and precipitation, observed RMW, and the radial–vertical B being the maximum wm at the base of the cloud at airflow through the eyewall of Hurricane Allen on 5 Aug 1980. radius rb. Here, DE represents the variation with height The slope of the cloudy region on the inside edge of the eyewall is based on radar minimum detectable signal analysis (10 dBZ), of the radius of maximum tangential wind (V␪)m and is inclined at an angle to the horizontal equal to or slightly aircraft altimeter readings, handheld photography, and observer ␸ ␺ notes. Here, A is the inner edge of the eyewall at the surface and different from , denoted by . It is assumed that the T is the top of the most developed cumulonimbus nearest to the radius of maximum wind varies linearly with height: cyclone center. Darker regions along BC denote the location of although fine-resolution model simulations indicate the largest radial and vertical velocities. The variation of the maxi- mum tangential velocity is along DE, with maximum low-level wind at D. (b) Schematic diagram illustrating the main features of (a) but with the addition of line EDЈ and angle ␺Ј.

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| Ј Ϫ | ␣ ϭ some departure from linearity of this feature (e.g., TABLE 1. Minimum and maximum rm rm for 10° and ␾ ϭ Zhang and Kieu, 2006); since we are only taking the 20° and for 75° and 60°. The range of h␷ is assumed to vary two end points in our calculation, this will not affect our between 5 and 7 km. final results. Max ␣ Min Max ␾ | Ј Ϫ | | Ј Ϫ | In Fig. 1a, the vertical line TEF passes through E, the (°) (°) rm rm (km) rm rm (km) maximum tangential wind (V␪)m at level E. Let the co- 75 10 1.0 1.4 ordinates of E and F be (rt, hV) and (rt, 0), respectively. 75 20 2.2 3.0 Because D is the maximum wind at the surface, the 60 10 1.1 1.8 60 20 2.0 4.3 coordinates of D are (rm, 0). Note also that rr, the ra- 45 10 2.1 3.0 dius of maximum precipitation rate in the lowest 2–3 45 20 3.7 8.0 km of the , measured from radar observa- tions, is on average greater by 3 km than rm but less of the CDO represents very warm temperature or part than rt (Jorgensen 1984b). According to Jorgensen of the eye, while the nearby green patch represents the (1984a,b), re, rm, and rr have all preferred locations rela- tive to each other and always occur in the same order, top of the middle-level stratiform cloud. In this case, so that the center of the outer edge of the semicircular dark blue arc is the cyclone center and the radius of the outer Ͻ Ͻ Ͻ ͑ ͒ re rm rr rt. 1 edge of the dark blue semicircular area is re. The near- Now, if we assume that angles TAF and EDF in Fig. est dark red cold band to the northeast of the cyclone 1a are nearly the same, then center, as indicated by the arrow, indicates a band of

ր ≅ ͑ Ϫ ͒ր͑ Ϫ ͒ ͑ ͒ highest cloud tops. Parameter rt is the shortest distance h␷ ht rt rm rt re , 2 from the cyclone center to the inner edge of the dark so that red cold band. Although the determination of rt, and ≅ ͓͑ Ϫ ր ͒ ϩ ͑ ր ͒ ͔ ͑ ͒ rm 1 h␷ ht rt h␷ ht re . 3 therefore of rm, is based on an arc of either the lowest However, if ␺ ␾, then an error will be introduced in brightness temperature or intense precipitation rate, it is assumed that the distance of the maximum wind from estimating rm from Eq. (3). The maximum error intro- duced from this inequality can be estimated from Fig. the cyclone center is the same around the cyclone; that 1b. In the case that angle EDF is ␺Ј, where ␺Јϭ(␾ Ϯ is, the deduced rm is axisymmetric. ␣ Ј Thus, if re, rt, and a mean value of h␷ /ht are known, ), then the radius of maximum wind would be rm, and Ј Ϫ ␾ϩ ␣ ␾ then the radius of maximum wind can be estimated (rm rm) would be equal to h␷[1/tan( ) – 1/tan ]. from Eq. (3). Hence, this method requires a well- There are not many evaluations of the parameters h␷, ␾, defined eye in the satellite imagery. We assume here and ␺. However, if we use an estimate of h␷ from dia- that h␷ /h and thereby the radius of maximum wind are grams published in Jorgensen (1984 a,b), then h␷ may t vary between 4 and 7 km. For a maximum value of ␣ of axisymmetric. The use of a mean value of h␷ /ht instead Ј Ϫ of one that varies from to storm introduces an 10°, the range of (rm rm) is between 1.0 and 1.4 km when ␾ ϭ 75° and between 1.1 and 1.8 km when ␾ ϭ error, but we show below that a good estimate of rm is 60°. For a maximum value of ␣ of 20°, the maximum obtained in spite of this approximation. Ј Ϫ value of (rm rm) would vary between 2.2 and 3.0 km when ␾ ϭ 75°, between 2.0 and 4.3 km when ␾ ϭ 60°, 3. Data and between 3.7 and 8.0 km when ␾ ϭ 45°. The esti- Table 2 gives observed values of parameters relevant Ј Ϫ ␾ mated ranges of (rm rm) for different values of and to the calculation of the RMW, observed RMW, and ␣ are given in Table 1. calculated RMW. Three series of data have been used

The parameter rt in Eq. (1) is the distance between to determine the mean value of (h␷ /ht) in Table 2. The the cyclone center and the coldest cloud-top tempera- first series of data gives re, the radius of the eye, and ture nearest to the cyclone center. It can be evaluated in RMW, the radius of maximum wind as observed by high-resolution IR satellite cloud data, or in TRMM reconnaissance flights in the North Atlantic, for three 85-GHz (85H) imagery. An example of a TRMM 85H storms: Anita, Allen, and Frederic. These are read off imagery is shown in Fig. 2. In this picture the TRMM from Figs. 4a–d in Jorgensen (1984a), which give plots imagery is overlaid over the corresponding Geostation- of tangential velocity and radar reflectivity versus ra- ary Operational Environmental Satellite (GOES) vis- dius. As shown in Fig. 1a, rt was assumed to be at the ible (VIS) imagery taken about 1 h earlier. In the tem- highest top nearest to the cyclone center of the 10-dBZ perature analysis, blue is very warm and dark red is radar reflectivity isoyet. very cold. The semicircular dark blue arc at the center The second series of data was obtained for two recent

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FIG. 2. TRMM image showing the horizontal distribution of the cloud-top brightness temperature around at 2052 UTC 27 Aug 2005. It is overlaid on a GOES-12 visible image at 1945 UTC on the same day. The dark red arc, indicated by the arrow, shows the location of the coldest

cloud-top temperature. The distance from the cyclone center to the inner edge of the arc is rt.

North Atlantic hurricanes, Katrina and Wilma, in Au- worth noting that for these surface wind analyses the gust 2005. For these storms, re was obtained from the winds were collected during a period of 2–3 h before 6-hourly National Hurricane Center hurricane warning and after the valid of the wind analysis. advisories (information online at http://www.nhc.noaa. As discussed above, rt was obtained from analyses of gov). Sometimes these were 2–3 h away from the time enhanced IR satellite cloud imagery or TRMM 85H of the satellite pictures when rt was determined, but imagery provided online (http://www.nrlmry.navy.mil/ such an independent assessment of re was preferred. tc_pages/tc_home.html). There were 16 sets of data for Based on aircraft reconnaissance, these estimates are these two hurricanes. typically accurate to about 5% (R. J. Deatherage 2008, The third series of data were obtained from archived personal communication). RMW was obtained from satellite data of North Atlantic tropical cyclones in the same Web site as for re, from the archived analyzed 2004. RMW was obtained as in the second series of surface wind field of the Atlantic Oceanographic and data, while re was estimated from imagery of colored Meteorological Laboratory (AOML) (Powell et al. analyses of enhanced IR satellite cloud data, from TMI 1998). Errors in RMW from aircraft reconnaissance can imagery of either brightness temperature or derived be as small as 1 km (J. L. Franklin 2008, personal com- rain rate, or from TRMM PR images. An example of munication), although the location of RMW can vary the enhanced IR imagery is shown in Fig. 3a, of TMI- rapidly with the changing structure of the storm. It is derived rain rate in Fig. 3b, while Fig. 4 shows a typical

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TABLE 2. Values of re, RMW (determined by NHC using wind data from reconnaissance flights, , buoys, and over a period of several hours), rt (estimated from IR or TRMM 85H), (h␷ /ht) calculated from Eq. (2), rm (calculated using a mean value of h␷ /ht of 0.6), and the difference between RMW and rm.

Ϫ Time and date of storm Obs re Obs RMW Obs rt Calculated rm |RMW rm| Hurricane observations (km) (km) (km) Obs h␷ /ht (km) (km) Anita 0300 UTC 2 Sep 1977 14 24.8 39.9 0.58 24.5 0.3 Frederic 0900 UTC 12 Sep 1979 28.8 31.5 43.5 0.81 30 1.5 Allen 1500 UTC 5 Aug 1980 27.6 48.7 57.5 0.49 40 8.7 8/2200 UTC 8 Aug 1980 8.5 9.7 12.4 0.69 10 0.3 N/A 26 33 45.5 0.64 34 1 Katrina 1841 UTC 26 Aug 2005 10 24 35 0.44 20 4 0732 UTC 28 Aug 2005 22 33 40 0.39 29 4 0815 UTC 29 Aug 2005 10 33 60 0.54 30 3 Wilma 0715 UTC 21 Oct 2005 28 31 32 0.25 30 1 1315 UTC 21 Oct 2005 28 35 50 0.68 37 2 1930 UTC 21 Oct 2005 23 29.5 36 0.5 28 1.5 2245 UTC 21 Oct 2005 23 30 40 0.5 28 2 2345 UTC 21 Oct 2005 23 33 48 0.6 30 3 0015 UTC 22 Oct 2005 23 26 28 0.4 25 1 0715 UTC 23 Oct 2005 18 54 96 0.54 49 5 0915 UTC 23 Oct 2005 18 55 96 0.61 56 1 0945 UTC 23 Oct 2005 18 55 120 0.63 59 4 1917 UTC 23 Oct 2005 41 65 98 0.58 63 2 2115 UTC 23 Oct 2005 41.5 61 100 0.67 65 4 0315 UTC 24 Oct 2005 41.5 61 100 0.67 65 4 0945 UTC 24 Oct 2005 60 74 120 0.77 84 10 Dennis 0620 UTC 7 Jul 2005 18 24 25 — 21 3 Charley 1900 UTC 12 Aug 2004 8 16 20 0.33 13 3 0430 UTC 13 Aug 2004 8 — 20 — 13 — Frances 0532 UTC 29 Aug 2004 16 22 29 0.54 21 1 1845 UTC 28 Aug 2004* 16 — 32 — 22 — 1020 UTC 30 Aug 2004* 16 — 47 — 21 — 1709 UTC 30 Aug 2004 6 34 65 0.73 29 5 1830 UTC 30 Aug 2004* 6 — 65 0.55 31 3 1752 UTC 31 Aug 2004 17 29 37 0.4 25 4 1735 UTC 31 Aug 2004* 17 — 34 0.3 24 5 Inner eye dissipating 0602 UTC 1 Sep 2004 14 36 65 0.57 34 2 1005 UTC 1 Sep 2004* 14 — 60 0.52 32 — 0645 UTC 2 Sep 2004 19 26 34 0.53 25 1 1045 UTC 2 Sep 2004* 19 — 30 — 23 — Ivan 0445 UTC 6 Sep 2004 6 11 16 0.5 10 1 0530 UTC 6 Sep 2004 6 — 13.5 0.33 9 2 1520 UTC 6 Sep 2004 6 — 13 — 10 — 0528 UTC 7 Sep 2004 8 13 32 0.79 18 5 0611 UTC 8 Sep 2004 6 13 25 0.63 13 0 0510 UTC 8 Sep 2004* 6 — 23 0.59 13 0 1500 UTC 8 Sep 2004* 8 — 22 — 13 — 1400 UTC 9 Sep 2004* 7 — 12 — 9 — 0500 UTC 10 Sep 2004* 7 — 28 — 15 — 1828 UTC 10 Sep 2004 8 9 12 0.75 10 1 0641 UTC 11 Sep 2004 6 17 37 0.42 19 2 0535 UTC 11 Sep 2004* 6 — 40 0.68 19 2 1350 UTC 11 Sep 2004* 6 — 28 — 15 — 0723 UTC 12 Sep 2004 6 16 20 0.29 12 4 0440 UTC 12 Sep 2004 6 — 22 0.38 12 4 1250 UTC 12 Sep 2004* 6 — 20 — 12 — 0520 UTC 13 Sep 2004* 10 — 25 — 16 — 1900 UTC 13 Sep 2004 16 29 32 0.19 23 6 0425 UTC 14 Sep 2004 12 — 31 — 20 — 1710 UTC 14 Sep 2004* 25 30 33 0.37 28 2 0752 UTC 15 Sep 2004 29 35 49 0.70 36 1 0510 UTC 15 Sep 2004 29 — 47.5 0.65 32 3 1850 UTC 15 Sep 2004* 30 33 41 0.73 28 5 0550 UTC 16 Sep 2004 30 — 35 — 32 — Mean ———0.55 — 2.8

* re was obtained from TMI or TRMM-PR imagery.

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FIG. 3. (a) An example of color-enhanced GOES IR imagery, suitable for

determining the size of re; shown is (2005). (b) An example of TMI imagery showing the derived distribution of precipitation rate around Hur- ricane Frances on 30 Aug 2004. Courtesy of NASA and the Aerospace Exploration Agency (JAXA).

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FIG. 4. An example of TRMM PR imagery showing the distribution of precipitation rate around on 30 Aug 2004. Courtesy of NASA and JAXA.

TRMM PR image. In the first picture, the dark brown indicates it is raining within the eye with a precipitation circular spot in the middle of the CDO is the warmest rate of about 7 mm hϪ1. The eye radius as measured spot or the eye of the cyclone, and a good estimate of re from this imagery was found to be 9 km. With rt, mea- can be made. Note, though, that the type of enhance- sured from a TRMM image, of 65 km, rm was estimated ment used in Fig. 3a is not suitable for determining rt to be 34 km. The nearest red or heavy to the because there is not enough temperature resolution in north east of the cyclone center was 52 km from the the red area surrounding the cyclone center to deter- cyclone center; that is rr, was 52 km. Thus, re, rm, rr, and mine the location of the nearest band of coldest cloud rt satisfy Eq. (1). tops from the cyclone center. Note when the eye is circular, a good estimate of re In Fig. 3b, the TMI imagery shows the distribution of can be made. However, when the eye is elliptical, for the precipitation rate around the cyclone: The blue re- instance if the ellipticity is caused by wavenumber 2 gion is rain free and the dark red areas are areas of Rossby waves (Montgomery and Kallenbach greatest precipitation rate. The semicircular red band in 1997), half of the longest axis has been taken as re. The the middle of the CDO is the eyewall. In this case also accuracy of determining re and rt on the computer the eyewall is not a complete circular band of clouds. screen is estimated to be Ϯ1.5 km. The center of the circle, of which the inner edge of this semicircular red band forms part, is the center of the 4. Results hurricane, and half of the diameter of this circle is re.In Columns 1–8 in Table 2 give, respectively, the name general, the eye is blue but in this case the yellow eye of the hurricane; the time and date of the AOML wind

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FIG. 5. Scatter diagram of measured RMW vs calculated rm. analysis or the time of the TMI rainfall intensity imag- It is worth noting that Hsu and Babin’s (2005) sug- ery; re (as determined by the U.S. National Hurricane gestion, namely that the RMW is equal to rt, the dis- Center, but when this was not available re was esti- tance between the coldest cloud-top temperature sur- mated from any available enhanced satellite cloud im- rounding the eye and the warmest temperature in the agery); RMW (obtained from AOML-analyzed surface eye, is not borne by the data presented in Table 2. wind fields); rt (obtained from either high-resolution There are only 19 out of 58 cases that have rt and RMW digitized IR data or TRMM 85H imagery); (h␷ /ht) cal- differing by 5 km or less. These occur when re and culated from Eq. (2); rm [calculated by using a mean RMW are small. For medium and large re and RMW, Ϫ value of 0.6 for (h␷ /ht) in Eq. (3)]; and |(RMW rm)|. the difference between rt and RMW is even greater, In Table 2 the with an asterisk indicate that re for varying between 15 and 35 km. that row was obtained from TRMM 85H or TRMM PR imagery. In the following analysis of the absolute error, 5. Discussion and conclusions cases when the difference between the time of the RMW and that of the TMI is greater than 4 h are not A simple technique for diagnosing the radius of considered. Despite the difficulties in estimating some maximum wind from satellite imagery has been pro- of the parameters and two rather large values of posed and tested against reconnaissance observations, Ϫ |(RMW rm)|, (10 km during Wilma at 0945 UTC 24 showing good agreement. The skill of this method is Oct 2005 was dissipating rapidly and 8.7 km when Allen comparable to or slightly better than that of Kossin et apparently had double eyewalls), the mean value of al. (2007), although here we analyze a smaller sample of Ϫ |(RMW rm)| is 2.8 km. Of the 58 cases considered, storms. In addition, our technique uses microwave im- Ϫ 50% of |(RMW rm)| are equal to or less than 2 km, agery, where the eye is often more clearly defined than 33% are between 3 and 4 km, and 17% are equal to or it is in IR images. One issue that should be examined in greater than 5 km. This compares with a mean absolute future work is the effect that vertical would error for the clear-eye method of Kossin et al. (2007) of have on the accuracy of this technique, as shear would

4.7 km, although our data sample is considerably displace rt. Lajoie (2007) argues that this displacement smaller than theirs. would also have a compensating effect on re and rm,

The calculated rm values from Eq. (3) have been plot- thus suggesting that this method might still give a rea- ted against the corresponding NHC-analyzed RMWs in sonable estimate of RMW, but this has yet to be tested Fig. 5. It is likely that some of the larger scattering of quantitatively. Another issue is the influence of asym- the data points is due to the fact that the times of the metries on the calculation of rm, as our method assumes wind and of the cloud data sometimes differ by up to 5 that derived quantities are axisymmetric. h. Two of the larger errors (Allen at 1500 UTC 5 Aug There are a number of potential applications of this

1980 and Wilma at 0945 UTC 24 Oct 2005) may have method. RMW or rm is an important parameter for been caused by the presence of double eyewalls, where diagnosing and forecasting the maximum wind of a our method would be expected to perform poorly. Oth- tropical cyclone. For example, studies estimating the erwise, the agreement is excellent. climatological impact of tropical cyclones on wave

Unauthenticated | Downloaded 09/27/21 06:36 PM UTC OCTOBER 2008 LAJOIEANDWALSH 1015 fields and usually have to assume a fixed of . Natl. Wea. Assoc. Electron. J., 2005-EJ3. [Avail- radius of maximum winds in ocean regions where there able online at http://www.nwas.org/ej/hsu/hsu_babin_2005.pdf.] are no routine observations of this quantity (e.g., Iguchi, T., T. Kozu, R. Meneghini, J. Awaka, and K. Okamoto, 2000: Rainprofiling algorithm for the TRMM Precipitation McInnes et al. 2003). Using the method outlined in this Radar. J. Appl. Meteor., 39, 2038–2052. paper, a diagnosed record of tropical cyclone structure Jorgensen, D. P., 1984a: Mesoscale and convective-scale charac- parameters can thereby be constructed for regions of teristics of mature hurricanes. Part I: General observations by the where such information has been difficult to research aircraft. J. Atmos. Sci., 41, 1268–1285. obtain to date. ——, 1984b: Mesoscale and convective-scale characteristics of ma- ture hurricanes. Part II: Inner-core structure of Hurricane There are other, more theoretical, applications. Allen (1980). J. Atmos. Sci., 41, 1287–1311. Lajoie (2007) has developed a simple mathematical Kossin, J. P., J. A. Knaff, H. I. Berger, D. C. Herndon, T. A. model that can determine the mean gradient-level wind Cram, C. S. Velden, R. J. Murnane, and J. D. Hawkins, 2007: averaged around the cyclone at a radius of 1° . Estimating hurricane wind structure in the absence of aircraft reconnaissance. Wea. Forecasting, 22, 89–101. It uses rm as a necessary parameter to determine the sustained mean maximum surface wind at r , and the Lajoie, F. A., 2007: A diagnostic study of three tropical cyclones in m the Australian region. Ph.D. thesis, University of Melbourne, radial distribution of the sustained mean surface wind 281 pp. along different orientations with respect to the direc- Lehner, S., A. Reppucci, and J. Schulz-Stellenfleth, 2006: Valida- tion of motion of the cyclone. The model has been used tion of parametric models by ENVISAT ASAR images. Ger- for three tropical cyclones operating in the Australian man Aerospace Centre (DLR), Wessling, Germany, 6 pp. region to successfully diagnose their maximum winds [Available from German Aerospace Centre (DLR) Oberp- faffenhofen 82234, Wessling, Germany.] when they pass over a meteorological station, the radial Lonfat, M., F. D. Marks, and S. S. Chen, 2004: Precipitation dis- distribution of the winds ahead and at the rear of the tribution in tropical cyclones using the Tropical Rainfall cyclone, as well as the time variation of the radial pro- Measuring Mission (TRMM) Microwave Imager: A global file of the mean surface wind. Details are contained in perspective. Mon. Wea. Rev., 132, 1645–1660. forthcoming publications. Marks, F. D., and R. A. Houze, 1987: Inner-core structure of Hur- ricane Alicia from airborne Doppler radar observations. J. Atmos. Sci., 44, 1296–1317. Acknowledgments. 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