A Diagnostic Study of the Intensity of Three Tropical Cyclones in the Australian Region

22 MONTHLY WEATHER REVIEW VOLUME 138

A Diagnostic Study of the Intensity of Three Tropical Cyclones in the Australian Region. Part II: An Analytic Method for Determining the Time Variation of the Intensity of a Tropical Cyclone*

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

(Manuscript received 20 November 2008, in ﬁnal form 14 May 2009)

ABSTRACT

The observed features discussed in Part I of this paper, regarding the intensiﬁcation and dissipation of Tropical Cyclone Kathy, have been integrated in a simple mathematical model that can produce a reliable 15– 30-h forecast of (i) the central surface pressure of a tropical cyclone, (ii) the sustained maximum surface wind and gust around the cyclone, (iii) the radial distribution of the sustained mean surface wind along different directions, and (iv) the time variation of the three intensity parameters previously mentioned. For three tropical cyclones in the Australian region that have some reliable ground truth data, the computed central surface pressure, the predicted maximum mean surface wind, and maximum gust were, respectively, within 63 hPa and 62ms21 of the observations. Since the model is only based on the circulation in the boundary layer and on the variation of the cloud structure in and around the cyclone, its accuracy strongly suggests that (i) the maximum wind is partly dependent on the large-scale environmental circulation within the boundary layer and partly on the size of the radius of maximum wind and (ii) that all factors that contribute one way or another to the intensity of a tropical cyclone act together to control the size of the eye radius and the central surface pressure.

1. Introduction intensity and radial wind profile. The model uses input parameters that are readily available and simple to use. The problem of forecasting tropical cyclone intensity The principles upon which the new model is based are and wind field remains an important issue in tropical further described here and then expanded in the en- meteorology. A review of recent work is contained in suing sections. Lajoie and Walsh (2010, hereafter Part I). Based on the Because different atmospheric processes are involved sequence of observed characteristic features during in different parts of a tropical cyclone, and also because intensification and dissipation of Tropical Cyclone the radial wind profiles in different regions have distinct Kathy in Part I, we have developed a mathematical physical features (Shea and Gray 1973), analysis of the model for forecasting all aspects of tropical cyclone circulation is considered in four distinct regions, as shown intensity. As discussed in Part I, it was suggested that in Fig. 1. Region 1 is the circular area of radius r , the eye the inflow of a band of moist, near-equatorial air into e radius. Region 2 is the annular area between r and r . the cyclone center was responsible for bursts of in- e m Region 3 is the area bounded between radii r and r , tensification of the storm. In addition, it was proposed m c the radius of the inner core of the cyclone or of the that the inflow angle of the band was related to the rate central dense overcast (CDO) region, which on average of intensification. The concepts are used here to de- is 18 latitude, as it was for Kathy. The two last regions velop an accurate numerical model of tropical cyclone contain the eyewall cloud, the maximum wind, most of the destructive winds, and very heavy precipitation. * Supplemental information related to this paper is available at the Region 4 is between rc and a radius of 48 latitude. Journals Online Web site: http://dx.doi.org/10.1175/2010MWR2876.s1. To start the analysis, an initial radial wind profile in region 4 a few hours before the start of intensification, or Corresponding author address: Kevin Walsh, School of Earth at t 5 0, is obtained from a streamline and isotach anal- Sciences, University of Melbourne, Parkville VIC, 3010 Australia. ysis. The initial radial surface pressure profile is then cal- E-mail: [email protected] culated to balance the wind profile. Because (i) the feeder

DOI: 10.1175/2009MWR2876.1

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In region 2, the wind is assumed to decrease linearly

from Vrm to zero at the cyclone center and the cyclostrophic wind equation is used to obtain pre, the surface pressure at re. Since surface pressure does not vary appreciably with radius in the eye of a cyclone (see rele-

vant barograms on 285–286 in Riehl (1954)), pre 5 pc, the central surface pressure.

Parameters rm and re are determined from the Trop- ical Rainfall Measuring Mission (TRMM) Microwave Imager (TMI; Simpson et al. 1988; Lonfat et al. 2004), TRMM Precipitation Radar (PR) data (Iguchi et al. 2000), or digitized high-resolution IR satellite cloud data (see Lajoie 2007; Lajoie and Walsh 2008). Finally, by using a reduction factor to convert gradient-level mean wind to mean surface winds, and the mean asymmetric FIG. 1. The inner and outer regions of a tropical cyclone. The wind distribution around a tropical cyclone, the hori- former is subdivided into the following: region 1, the eye where 0 # zontal distribution of the sustained mean surface wind as r # re; region 2 where re , r , rm (rm being the radius of maximum well as the maximum gust around the cyclone can be wind); region 3 where rm # r # rc (rc being the external radius of the estimated. CDO); and region 4, the outer region where r , r # 48 latitude. c Section 2 of this paper details the mathematical formulation of the prediction model. Section 3 gives results band originates at about 440 km from the cyclone center, of hindcast predictions of intensity changes for Tropical (ii) only moist near-equatorial air spirals through the Cyclone Kathy using the model. Section 4 shows calcu- feeder band into the cyclone core and no cool dry trade air lations of the estimated radial wind profile in Kathy. participates in the cyclonic circulation during intensifica- Section 5 gives hindcast intensity predictions for two tion, and (iii) the convective cloud suppression around the other cyclones, section 6 discusses issues associated with cyclone during the period of intense convection in the the intensification rate of cyclones, and section 7 pro- CDO is observed to extend to about that distance, it is vides concluding remarks. assumed that all physical processes that influence the cyclone intensity take place within 440 km of the cyclone 2. Model formulation center and that only the cyclonic circulation starting from 440 km to the north and northeast of the cyclone center a. Region 4 (rc # r # r4) needs to be taken into consideration to determine the 1) CIRCULATION ANALYSIS IN REGION 4 change in its intensity or in its intensification rate. An energy equation is then applied to the circulation Streamlines around stationary axisymmetric tropical in region 4. It is shown that because of the sudden cyclones are logarithmic spirals (see Abdullah 1966; change of a4, the angle of inflow at a radius of 48 latitude, Lahiri 1981; Wong et al. 2007). According to Senn and to the north and east of the cyclone at t 5 0, gradient- Hiser (1957, 1959), the trajectories of air parcels are level winds subsequently increase at all radii in region 4 equiangular logarithmic spirals far from the cyclone (48 latitude . r $ 18 latitude). On assuming that air center, while close to the center the angle of inflow de- trajectories are logarithmic spirals, the distances trav- creases with decreasing radius. A similar assumption is eled by air parcels between selected radii can be de- used here. In region 4, the angle of inflow a (see Fig. 2), termined, and from the radial wind profile, Vrc the wind in the northerly airstream to the north and northeast of at r 5 rc or 18 latitude and the time at which air parcels the cyclone, which is assumed to be constant between leaving r 5 48 latitude reaches rc can also be determined. radii 48 and 28 latitude and varies as shown in Fig. 2 for In region 3, a theoretically determined gradient-level smaller radii. The maximum angle of inflow is 208 before wind profile, particularly developed for that region by intensification, 42.58 and later 338 during intensification, Riehl (1954, 1963) and empirically documented by Shea and 208 during dissipation as observed in Fig. 5 of Part I. and Gray (1973), is then used to determine the gradient- The angle of inflow a is assumed to decrease linearly to level winds at different selected radii and the maximum 208 for 28 latitude $ r $ 18 latitude and to decrease to 108 wind Vrm at rm. The gradient wind equation is then used for 18 latitude . r $ 08 latitude. The last assumption is to obtain the surface pressure gradient and the surface made to simplify the computations of pc without greatly pressure at selected radii and at rm. affecting the results.

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The solid curves ending with an arrow represent the streamlines. AB is an arc of a circle of radius r. The quantity P is the location of an air parcel moving along the s direction with a speed V. Vector n is normal to the s direction, u is the angle POX, and O is the cyclone center. Following Malkus (1958, 1962) the equations of motion of an air parcel at P in natural coordinates, assuming large-scale two-dimensional ﬂow, can be written as dV ›V ›V w›V 1 ›p 5 1 V 1 5 À 1 F, dt ›t ›s ›z r ›s (1) V2 À1 ›p 1 ›p 1 fV5 5 cosa , (2) R r ›n r ›r r

FIG. 2. Radial variation of the angle of inﬂow a at different times. where V is the horizontal wind at the gradient level along the s direction, t is time, F is the frictional accel- It is further assumed, following Malkus et al. (1961), eration, R is the radius of curvature of the air trajecto- Malkus (1962), Gentry (1984), and also Lucas et al. (1994), ries, and f is the Coriolis parameter. The following that 1%–5% of air parcels in region 4 will rise in convective assumptions are used in Eq. (1): updrafts, and the rest will move on a quasi-horizontal trajectory to reach rc, the outside edge of the CDO. 1) In the expansion of dV/dt in Eq. (1), the term ›V/›t is The natural and polar coordinate systems used in the the local rate of change of V. For ›V/›t to change, the analysis of the circulation in region 4 are shown in Fig. 3. surface pressure gradient in region 4 must change

FIG. 3. Natural and polar coordinate system used to analyze the flow of near-equatorial air toward the cyclone. The curves are logarithmic spirals representing air trajectories. Air parcels at point P move along the s direction. The n is at right angle to s, O is the cyclone center, u is the angle that OP makes with a fixed arbitrary axis, and a is the angle of inflow (angle between s and the tangent at P to the arc of a circle of radius OP).

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FIG. 4. The two logarithmic spirals T1 and T2 represent trajectories of near-equatorial air with different angles of inﬂow at r 5 48 latitude distance. The DS is the distance between selected radii along the trajectory. Here Vr and pr are the wind speed and surface pressure, respectively, at r. Subscript 1 or 2 refers to T1 or T2.

with time, but we have shown in Fig. 14 of Part I that The reason for this assumption is because (i) only the surface pressure at Centre Island (assumed to be moist near-equatorial air originating at 440 km to the the same at any radius r in region 4) was constant north and northeast of the cyclone spirals through for about 9–12 h before a sudden surface pressure fall feeder bands into the cyclone core (no cool dry trade that lasted about 3 h. Thus, ›V/›t 5 0 for the 9 h, while air participates in the cyclonic circulation during in- during the following 3 h it will be shown that the tensification) and (ii) the convective cloud suppres- change of V is brought about by the term V(›V/›s). sion around the cyclone during the period of intense 2) Since we shall be using Eq. (1) only in region 4 and convection in the CDO is observed to extend to only at the gradient level (i.e., at the height of max- about that distance. imum wind, where ›V/›z vanishes), neglecting of the 6) The surface pressure field and hence the surface pre- term w›V/›z is entirely justified. ssure gradient outside the CDO remain constant while 3) The influence of the ocean–atmosphere surface drag ‘‘near-equatorial air’’ moves from r 5 48 latitude to

is negligible at the gradient level. r 5 rc or 18 latitude and changes when it moves from rc 4) The wind direction is almost constant with height near to the eyewall. the gradient level and the frictional force that is mainly Equation (1) therefore reduces simply to due to the vertical eddy transport of momentum by turbulence and convective elements is mostly along dV ›V 1 ›p 5 V 5 À 1 F. (1a) the s direction and negligible along the n direction. dt ›s r ›s 5) The surface- and gradient-level wind and the surface pressure at r 5 48 latitude distance from the cyclone The symbols used in the analysis are illustrated in Fig. 4,

center remain constant during the evolution of the which shows two trajectories: T1 with a4 of 208 and T2 cyclone. with a4 of 42.58. The former is the trajectory of air

Unauthenticated | Downloaded 09/29/21 04:37 AM UTC 26 MONTHLY WEATHER REVIEW VOLUME 138 parcels when Kathy was in a steady nonintensifying state Friction at the gradient level is due to turbulent mixing and the latter when it had started to intensify. Suppose and thermal updrafts at that level and can be regarded as 2 that at time t 5 0 a parcel of air is at point Po1 in Fig. 4 on a drag force, which is proportional to (V 2 ub) , where ub radius ro, where the surface pressure is po1, and where is the horizontal component of the wind inside and at the the air parcel is moving along trajectory T1 with an initial base of the updrafts (Malkus 1952). Since on the one velocity Vo1. Suffix ‘‘o’’ denotes the initial position or hand, air that is entrained from below the gradient level initial values of p or V. Suppose also that in time dt,theair tends to conserve its horizontal momentum (Austin and parcel has moved along T1 through a distance Ds1 and has Houze 1973), and on the other, surface winds are lighter reached the point Pf1 on radius rf (5r 2 dr) where the and proportional to V (actually 0.8V, see section 2b), 2 2 surface pressure is pf1 and where its speed is now Vf1. then F is proportional to V and can be written F 5 kV , Suffix f denotes the final position or the final values of p where k is a constant. If we assume that in a small finite and V after dt. Suffix 1 denotes that the air parcel has been distance Ds1, V increases linearly, then V 5 Vo 1 ls and moving on trajectory T1. If the air parcel had been ð ð Ds1 Ds1 moving on trajectory T its final position after dt would be 2 2 Fds5 k(Vo 1 ls) ds, 0 0 Pf2 (see Fig. 4) and the surface pressure and wind speed would be pf2 and Vf2. Now if the air density in region 4 is where l 5 (Vf 2 Vo)/Ds1. The friction term then be- assumed to remain constant and Eq. (1a) is integrated comes (for T1) with respect to s between s 5 0ands 5Ds1,weobtain ð ð Ds Ds 1 kDs 1 1 1 1 2 2 2 2 Fds5 (Vo1 1 Vf1 1 Vf1Vo1), (Vf1 À Vo1) 5 (po1 À pf1) À Fds. (3) 0 3 2 r 0

Equation (3) is an energy equation similar to that derived and Eq. (3) becomes (for T1) by Haltiner and Martin (1957) except for the vertical motion term that is neglected here for reasons discussed 2 2 2 2k Vf1 À Vo1 5 (po1 À pf1) À Ds1 above. Physically, Eq. (3) states that the increase in r 3 kinetic energy of the air parcel is simply the difference 2 2 3 (Vo1 1 Vf1 1 Vo1Vf1). (4) between the work done by the pressure gradient force and the energy dissipated by friction. In other words, Similarly, air parcels moving from point Po along tra- when the system is in a steady-state, nonintensifying jectory T2 instead of along trajectory T1, would arrive at condition, that is when a4 5 208, the work done by the point Pf2 with a velocity Vf2 given by the pressure force is then just enough to balance the energy dissipated by friction and to maintain the ex- 2 2 2 2k Vf2 À Vo2 5 (po2 À pf2) À Ds2 isting wind ﬁeld. It is worth noting that Eq. (3) is used in r 3 region 4 only. In regions 2 and 3, because of the many 2 2 3 (Vo2 1 Vf2 1 Vo2Vf2), (5) complex atmospheric processes that take place there and that affect the wind speed we will use the radial where Ds2 is the distance between points Po2 and Pf2 mean wind proﬁles that have been determined by other along T2. By substituting for k in Eqs. (4) and (5), we investigators. obtain the following:

2 Ds (V2 1 V2 1 V V ) V2 À V2 À (p À p ) 2 2 o2 f2 o2 f2 f1 o1 r o1 f1 V2 5 V2 1 (p À p ) 1 , (6) f2 o2 r o2 f2 2 2 Ds1(Vo1 1 Vf1 1 Vo1Vf1)

2 2 2 ysis prior to the start of intensiﬁcation. As discussed in Vf1 À Vo1 À (p À p ) r o1 f1 section 2a(2), Ds1 and Ds2 can also be evaluated and are Let A 5 . (7) 2 2 constants for any particular ro and rf. Equation (6) can (Vo1 1 Vf1 1 V V ) o1 f1 now be written as All terms in A are values of meteorological parameters 2 2 2 Ds2 on trajectory T1 (Fig. 4). They are constants and can be Vf2 5 Vo2 1 (po2 À pf2) 1 A r Ds1 determined for any particular values of ro and rf from 2 2 a gradient-level wind analysis or a surface pressure anal- 3 (Vo2 1 Vf2 1 Vo2Vf2). (8)

Unauthenticated | Downloaded 09/29/21 04:37 AM UTC JANUARY 2010 L A J O I E A N D W A L S H 27 2 3 Equation (8) is used to calculate Vf2 for the following 1 6tan aro7 selected radii 3.58,3.08,2.58,2.08, ...,18 latitude. At 1 6 2 7 Ds 5 loge4 5, (13) radius r2 5 18 latitude, Vf2 5 Vrc, the gradient-level m 1 tan a wind at the edge of the CDO. Because all parcels of 2 rf air originating at a radius of 48 latitude between north r and east-northeast of the cyclone (see Fig. 4) with the R 5 , (14) same original speed and angle of inflow, and travel the (1 À rm tanar) cosar same distance Ds2, they arrive at the same time at rf2 5 18 latitude, so that Vrc is actually the mean wind aver- where aro and arf are the angle of inflow at ro and at rf, aged around the cyclone at r 5 rc 5 18 latitude. Here Vrc respectively. is an important parameter as there is a good relationship between it and the mean winds at different radii in 3) TIME INTERVAL FOR AN AIR PARCEL TO region 3 (Hughes 1952; Riehl 1954, 1963; Shea 1972). MOVE BETWEEN TWO SELECTED RADII Before discussing the method used for computing Vf2, The time interval Dt for an air parcel to move onto however, it is necessary to describe the method used to trajectory T2 from an initial radius ro to a final radius rf obtain values of other parameters. where the initial and final velocities are Vo2 and Vf2,is given by 2) DETERMINATION OF DS AND R As stated before, the trajectories of parcels of equa- (2Ds ) Dt 5 2 . (15) torial air are assumed to be logarithmic spirals with (Vo2 1 Vf2) a radial profile of a, as specified in Fig. 2. The basic equation for a logarithmic spiral is b. Region 3 (r # r # r ) m c 1 ›r 1) THE RADIAL PROFILE OF TANGENTIAL 5 tanar 5 tanmr, r ›u WIND IN REGION 3 where u is the angle between r and a fixed direction and To obtain the radial wind profile for a tropical cyclone m 5 ›ar /›r (see Fig. B2 in Appendix B in the supple- many investigators have assumed an hyperbolic radial mental material). It is shown in Appendix B (in the profile of surface pressure from some 600 km or more supplemental material) that the distance Ds along the to rm and then use the gradient wind equation to obtain trajectory from an initial radius ro to a final radius rf and the radial wind profile (see e.g., Knaff and Zehr 2007; the radius of curvature R at any point on the trajectory, Holland 1980, 2008). However, because the radial wind required when using Eq. (2), are given by profile and the atmospheric processes involved in region 3 are different from those in other regions (Shea and Gray "# ð 2 1/2 r f 1973), we need to use an equation that has been de- du 2 dr Ds 5 r 1 dr, (9) veloped for that particular region. r dr du o By assuming that potential vorticity is conserved in "# 3/2 the lower inflow layer of the atmosphere and conser- dr 2 r2 1 vation of angular momentum in the outflow layer aloft, du Riehl (1954, 1963) has shown that if Vur is the mean R 5 2 ! 3 . (10) 2 tangential wind at radius r, averaged around the vor- d2r dr 4r2 À r 1 2 5 tex, then du2 du x Vur 3 r 5 a constant, for region 3, For equiangular spirals with an angle of inflow a: where rm # r , rc (16) (r À r ) Ds 5 o f , (11) and x 5 0.5 for surface winds. Studies of the wind sina structure in the inner core of tropical cyclones by r Hughes (1952) and Shea (1972) lead to a similar con- R 5 . (12) cosa clusion. Also, using a large number of individual legs of low-level aircraft flight data in the North Atlantic from

For spiral trajectories whose angle of inﬂow ar decreases r 5 75 km to r 5 rm, Shea and Gray (1973) obtained linearly with r: a mean value for x of 0.47 with a standard deviation of

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0.3. They attributed the rather large standard deviation radial and azimuthal distribution of the mean 900-hPa to ‘‘the large variability of asymmetries and of sub- wind speeds, averaged over a large number of tropical stantial day-to-day and even hour-to-hour changes in cyclones, such as in Fig. 20 of Shea and Gray (1973). The intensity of individual storms.’’ In the current study, main asymmetry appears to be restricted to within 40 km Eq. (16) will be used to determine the radial mean wind from the radius of maximum wind. In the supplemental distribution at gradient-level with x 5 0.5. Asymmetries material, Table S1 summarizes their results. The average of speeds in region 3 will be discussed in section 3. winds were obtained along different directions relative to

If the mean wind at any point on radius r is Vr and the the direction of motion. The ratio of the mean wind at angle of inflow ar is constant at any radius r, then Vur 5 any point to the mean wind averaged around the cyclone Vr cosar and Eq. (16) becomes is given in column 4 of Table S1 (see the supplement). This ratio, multiplied by 0.8, gives the ratio required to 0.5 cosarc rc convert the gradient-level mean wind to the sustained Vr 5 Vrc , (17) cosar r mean surface wind, averaged around the cyclone, and is listed in column 5. where Vrc is the mean wind at radius rc, the radius of the URFACE WIND GUSTS CDO; arc is the angle of inflow at radius rc in particular, 4) S Vrm; and the mean wind at the radius of maximum wind The ratio of the peak wind gust to the mean sustained rm is given by wind speed at the surface varies according to the nature of the terrain, being 1.25:1 for coastal regions or for 0.5 cosarc rc small islands (Atkinson 1974; Padya 1976) and 1.41:1 for Vrm 5 Vrc , (18) cosarm rm inland stations. Thus, the maximum gust ahead of or to the left of the where arm is the angle of inflow at rm. Note that Vrm is direction of motion can be obtained by multiplying the dependent on arc, Vrc, rc, and rm. mean gradient-level wind at rm by the relevant factor in Table S1 (see the supplement) and by 1.25 or 1.41.

2) SURFACE WIND VS c. Surface pressure The variable Vs, the 10-min mean wind speed at 10 m above the ocean surface, is related to V, the mean gradient- 1) SURFACE PRESSURE IN REGIONS 3 AND 4 level wind speed, by V 5 kV. The quantity k is de- s Using a large number of low-level aircraft data, pendent on boundary layer stability conditions and Willoughby (1990) found that the gradient wind de- varies between 0.7 and 0.9 (Gray 1972; Garratt 1977; termined from the gradient of isobaric height is a good Powell 1980, 1982; Georgiou 1985; Franklin et al. 2003). approximation to the azimuthally averaged tangential Gray (1972) used a factor of 0.8 to relate 900-hPa winds wind in the free atmosphere, on the inner side of r as to V . This value will be used in this study to convert the m s well as on the outer side up to 150 km from the cyclone mean gradient-level wind to the mean surface wind av- center. We therefore use the gradient wind equation: eraged around the cyclone. ›p rV V 3) WIND SPEED ASYMMETRY 5 f2 f2 1 f (19) ›r cosaf2 R Tropical cyclones in the Southern Hemisphere have their maximum winds to the left and their minimum to to obtain the surface pressure gradient that is required to the right (Holland 1980). The asymmetry is only partly balance the mean tangential gradient-level wind in re- due to cyclone motion, the distribution of relative gions 3 and 4. In Eq. (19), af2 is the angle of inflow at final winds around a tropical cyclone also being asymmetric radius rf2 and R is the radius of curvature of the trajec- (Sherman 1956; Sheets 1982; Weatherford and Gray tory T2 at rf2. 1988; Knaff and Zehr 2007). Factors that can contrib- The surface pressure is assumed to remain constant at ute to wind speed asymmetries are asymmetric con- r 5 48 latitude during the life of the cyclone and the vection (Malkus et al. 1961; Shapiro 1983; Schubert surface pressure at r 5 3.58 latitude is obtained by taking et al. 1999), the development of low-level jets (Kepert into account the mean pressure gradient between r 5 48 and Wang 2001), and nonlinear b effects (Fiorino and and r 5 3.58 latitude, and so on for the other radii. In this Elsberry 1989; Wang and Holland 1996). To take account way a new surface pressure field is obtained from r 5 48 of the wind asymmetry around a tropical cyclone, we latitude in region 4 to rm the radius of maximum wind in have assumed that it is similar to that obtained from the region 3.

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2) THE CENTRAL SURFACE PRESSURE PC from a reconnaissance aircraft, or again when it passes over a meteorological station if the time interval of the Depperman (1947) and Shea and Gray (1973) sug- cloudless sky and the translation speed of the cyclone gested that the tangential wind speed decreases linearly are known. with decreasing radius between r and r . Since V 5 0 m e In infrared and visible satellite cloud images the eye is at r 5 0, then (V ) /r 5 (V /r), so that V at any u rm m ur ur sometimes partially and sometimes almost fully covered radius r in region 2 is given by V 5 (V ) 3 (r/r ) for ur u rm m by low clouds, or by convective clouds that are associated r # r , r , where (V ) is the tangential wind speed at e m u rm with vortex Rossby waves (VRW) that travel along the radius r . Noting that (V ) 5 V cosa ,(a is the m u rm rm rm rm inner edge of the eyewall (Montgomery and Kallenbach angle of inﬂow at r ) and the fact that the ﬂow inside the m 1997; Wang and Wu 2004). With one VRW the eye has radius of maximum wind is in solid rotation and in cy- an irregular shape and part of the circular eye that is clostrophic balance (Emanuel 1995) we have for any visible allows the estimation of the eye diameter. When particular value of V : rm there are two VRWs that are opposite to one another ! because of the effect of wavenumber 2, the eye appears ›p rV2 cos2a 5 rm rm to be elliptical. In this case the eye diameter is more À 2 r. (19a) ›r rm likely to be the longer axis of the ellipse. Because the eye slopes outward with height, a correction has to be ap-

Since the surface pressure inside the eye does not vary plied when estimating re at sea level from visible and IR appreciably with r (Riehl 1954), we can write for most satellite cloud data. For the intense Tropical Cyclone practical purposes pc 5 pe,(pe being the surface pressure Kathy, the correction was made by assuming that the at re). Here Dp, the pressure difference between rm and slope was 758, as was observed by Jorgensen (1984) for re, can be obtained by integrating the cyclostrophic intense hurricanes (see Lajoie 2007). The values of re for equation; thus, Kathy are shown in Table S2 (see the supplement). ð ! r 2 2 4) THE PSEUDOEYE e rV cos a Dp 5 À rm rm rdr. (20) r2 According to Eq. (21), pc, the central surface pressure, rm m can be calculated if re and rm are known. Sometimes rm can be estimated (Kossin et al. 2007; Lajoie and Walsh

If for short periods of time Vrm, arm and rm are regarded 2008), but re cannot be determined either because the as constants (actually their time changes are very small, eye has not yet developed or the eye is not visible in even in rapid intensification; as discussed by Kaplan and satellite cloud images because of the cirrus shield over it. 21 21 DeMaria 2003), Vrm changed by about 0.7 m s h then Under these circumstances, we have to use a pseudo eye radius re* to estimate pc. To determine re*, a sample of r2 North Atlantic data of r (obtained from 6-hourly hur- Dp is À0.5(rV2 cos2a )1À e and e rm rm r2 ricane warning advisories from National Hurricane m r2 Center) and the radius of maximum wind (RMW) ob- p 5 p À 0.5(rV2 cos2a )1À e , (21) c rm rm rm r2 tained from the archived analyzed surface wind fields of m the Atlantic Oceanographic and Meteorological Labo- ratory (AOML), was used to prepare Table S3 (see the where p is the surface pressure at r . Equation (21) rm m supplement). Five pairs of r and RMW were associated indicates that p depends on four parameters: p , r , r , e c rm e m with cyclones having double eye walls and were not in- and V . The latter also depends on V and r . The rm rc c cluded in Table S3 (see the supplement). larger V and the smaller r , the smaller is p . Also for rm e c Table S3 (see the supplement) indicates the possible the same V and r , the larger r is, the smaller is p . rm e m c range of r* for small ranges of RMW. The concept of using Thus, to determine the tropical cyclone intensity it is e r* is particularly useful to determine p in the early stages imperative to use an accurate value of r and r . A tech- e c e m of intensification when the eye has not yet developed and nique to determine r is discussed by Lajoie (2007) and m when the surface pressure gradient is still small. Lajoie and Walsh (2008). For Kathy the values of rm at different times are given in Table S2 (see the supplement). 3. The physical processes of intensification of 3) THE EYE RADIUS Tropical Cyclone Kathy

The radius of the eye re can be determined accurately In the previous section, all the observed features dis- when the eye is within radar range or when it is observed cussed in Part I have been integrated into a simple

Unauthenticated | Downloaded 09/29/21 04:37 AM UTC 30 MONTHLY WEATHER REVIEW VOLUME 138 mathematical model to determine the time variation of concentric circles with their centers at the cyclone center, the wind speed and of the surface pressure at various then initially pf1 5 pf2,sothat(po1 2 pf1) 5 (po2 2 pf2). selected radii, including Vrm and pc at rm and re, re- The aim of the computations is to use Eq. (8) to de- spectively. The computation procedures and the results termine Vf2 at each selected radius in region 4 up to Vrc of the computations are discussed below. at rf2 5 18 latitude. Since all other parameters in Eq. (8) are constants or can be theoretically determined, Eq. (8) a. Initial environmental wind and surface pressure has become a simple quadratic equation with constant 2 profiles for Kathy coefficients as a Vf2 1 bVf2 1 c 5 0: The initial wind speeds for the following selected radii Ds where a 5 1 À A 2 , (22) 4.08,3.58,3.08, ...,18 latitude were estimated along Ds streamlines in the northeast sector of the cyclone from 1 the objective gradient-level streamline and isotach anal- Ds2 ysis of 1100 UTC 19 March 1984, just before Kathy b 5ÀA , (23) Ds1 crossed the coast and when air parcels were moving along trajectory T1. These winds are given in column 2 2 Ds c 5 V2 1 p p 1 A 2 V2 of Table S4 (see the supplement). The surface pressure À o2 ( o2 À f2) o2 . r Ds1 gradients that balance the observed winds were calculated (24) for each radius by using Eq. (19) in the following form: Given the change of the angle of inflow at the start of 1.293Vr Vr (Dp)r 5 1 0.0352 , (19b) intensification, the initial wind speeds and surface pres- cosar R sure values at selected radii in region 4, Vf2 at each of these radii increased to satisfy Eq. (8). They are shown in where (Dp)r is the pressure gradient in hectopascals the last column of Table S5(a) (see the supplement). per 18 latitude distance, Vr is in meters per second, R is in kilometers, r 5 1.15 kg m23,andf inEq.(19)isthe 2) COMPUTATIONS OF FORECAST Coriolis acceleration at 158 latitude. WINDS IN REGION 3 The surface pressure for 3.58 latitude was determined using the surface pressure at 48 latitude (1006.6 hPa) The radial wind profile of the mean tangential winds is and the mean surface pressure gradient between 48 and discussed in section 3a. Using Eqs. (17), (18), and the 21 3.58 latitude [0.55 and 0.6 hPa (18 latitude) ]incolumn3 newly computed mean gradient-level wind Vrc at r 5 of Table S4 (see the supplement), where the surface 18 latitude, and taking into account of the variation of pressure at 3.58 latitude is 1006.6 2 1/4(0.55 1 0.60). In the angle of inflow (see Fig. 2), the mean gradient-level the same way the surface pressure at other radii were winds averaged around the cyclone at selected radii in determined. The surface pressure and pressure gradi- region 3 can be computed. They are shown in the three ent at each radius are also given in Table S4 (see the last lines of Column E of Table S5(b) (see the supple- supplement). ment). The maximum mean gradient-level wind is that

calculated for rm. b. Computations of parameters during the lifetime of 3) THE FORECAST CENTRAL SURFACE PRESSURE the ﬁrst feeder band First, the time intervals taken for an air parcel to move 1) COMPUTATIONS OF FORECAST from one radius to the next are determined from Eqs. WINDS IN REGION 4 (11), (13), and (15). They are shown in column K of Computations of the winds during the evolution of the Table S5(b) (see the supplement). Since intensiﬁcation

first feeder band are first made for the segment from ro 5 started at 2100 UTC 19 March, the times at which the 48 latitude to rf1 5 rf2 5 3.58 latitude (see Fig. 4). Ini- first parcels of moist near-equatorial air reached differ- tially, ro1 5 ro2, po1 5 po2 and Vo1 5 Vo2. A description ent rf2s are given in column L of Table S5(b) (see the of the terms rf1, rf2 .... Vo2 is given in Appendix A (see supplement). They reached the edge of the CDO (radius the supplement). As observed in section 4 of Part I, 18 latitude) at 0930 UTC 20 March (to the nearest half the radial surface pressure remained constant in region 4 hour). Three hours later (i.e., at 1230 UTC 20 March), (48 latitude $ r $ 18 latitude) while near-equatorial air moist near-equatorial air has reached the eyewall (see moved from r 5 48 to 18 latitude. After all, this is quite section 3a(v) of Part I) at a radius of 0.38 latitude, where logical since no atmospheric process had occurred yet to the maximum wind around the depression was 35.3 m s21 cause any surface pressure change. Now if the isobars are [see column B Table S5(b) in the supplement].

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Then, the radial surface pressure profile is determined The intensification rate stayed 25 hPa day21 for about using Eq. (19b) to calculate the surface pressure gradi- 2 h before decreasing almost to its original value (see ents that are required to balance the newly computed Fig. 13 of Part I). It is likely that this decrease of in- winds in both regions 4 and 3. Assuming that the surface tensification rate is due to the advection into the CDO of pressure at r 5 48 latitude was outside the cyclone in- ‘‘subsidence dried air’’ in the lower and middle layers of fluence and therefore remained constant during the life the troposphere [see section 3a(ii) of Part I]. of the cyclone at 1006.6 hPa, and using the computed surface pressure gradients, the surface pressure at all c. Computations of parameters during the lifetimes of radii are determined as discussed in the last paragraph of other feeder bands section 3a. They are shown in column I of Table S5(b) (see the supplement). The forecast central surface pres- Tables of detailed computations for the four feeder sure is the computed pressure at re. bands can be inspected in Lajoie (2007). Table S6 (see Since at 1230 UTC 20 March the depression had not the supplement) lists the computed surface pressure, yet developed an eye, to calculate pc, the central surface sustained maximum mean surface winds (0.91 times the pressure, we use re*, (the radius of a pseudo eye, see mean gradient-level wind, see Table S1 in the supple- section 2 above) to compute pc. According to Table S3 ment) at different times for the four feeder bands. (see the supplement), for rm 5 0.38 latitude, re* varies Air parcels starting from ro 5 2.58 latitude at 1230 UTC between 0.28 and 0.278 latitude. Plugging these values in 20 March for the second feeder band with Vo2 5 21 Eq. (21), we obtain pc 5 (987 6 1) hPa at 1230 UTC 11.3 m s [see Table S5(a) in the supplement], reached 20 March. The Bureau, using the Dvorak (1975, 1984) the 18 latitude radius at about 1800 UTC 20 March. technique, estimated that the central surface pressure at Three hours later (2100 UTC 20 March), they reached 1230 UTC 20 March was 989.5 hPa. a radius of 0.258 latitude when the maximum surface mean wind was 36.8 m s21. At that time the eye had not 4) THE CHANGE IN THE INTENSIFICATION RATE yet developed and we have to use re* of Table S3 (see the The time variation of the computed central surface supplement) to obtain pc. According to that table, for pressure and of the rate of intensification of Kathy is a RMW (rm) of 0.258 latitude re* varies between 0.098 and shown in Figs. 3 and 13 of Part I, respectively. Between 0.238 latitude. Plugging these values in Eq. (21), we ob-

2100 UTC 19 March and 0900 UTC 20 March (i.e., be- tain a computed pc of 980 6 3 hPa. The Bureau’s esti- fore the arrival of near-equatorial air in the CDO), the mate of the central surface pressure at that time was intensification rate was 10 hPa day21.Sometimebe- 981 hPa. tween 0900 and 1200 UTC 20 March the intensification During the lifetime of the second feeder band, moist 21 rate suddenly increased to 25 hPa day .Figure13of near-equatorial air parcels starting from ro 5 48 latitude Part I shows that this change of intensification rate reached the edge of the CDO at 0000 UTC 21 March and occurred in the 3 h when the high-level cirrostratus rm (0.168 latitude, see Table S2 in the supplement) at area was increasing; that is, when the convective ac- about 0330 UTC 21 March (to the nearest half hour). tivity in the CDO was more intense and also when the With re 5 0.088 latitude (see Table S2 in the supplement) surface pressure at Centre Island (outside the CDO) the computed Vrm at maximum surface wind and pc were suddenly fell by 0.6 hPa below what it would have been 46.8 m s21 and 961 hPa, respectively. The Bureau’s es- if the fall was due to diurnal variation alone. Similar or timate of pc at that time was 971 hPa. larger sudden falls have been observed for other storms Note that the computed period in which moist near- (Lajoie 2007). equatorial air moved into region 3 (from 0000 to 0330 UTC It is suggested that these sudden surface pressure falls 21 March) is very close to the period (0200 and 0500 UTC inside and outside of the CDO were due partly to the 21 March) when an observed sudden small surface pres- rather sudden burst of extra amount of condensational sure fall occurred at Centre Island, about 325 km from heating in the upper level that could extend in the out- the cyclone center (see Fig. 14 and Table 2 of Part I). flow layer to about 500 km from the cyclone center (see Again, within that 3-h period the instantaneous inten- Frank 1977; Anthes 1982). This surface pressure fall at sification rate (solid curve in Fig. 13 of Part I), based on about 1230 UTC 20 March produces a new radial surface the time variation of the computed central surface pres- pressure profile to balance and maintain the gradient- sure, increased suddenly from 25 to 100 hPa day21.It level winds in region 4. This new radial surface pressure stayed at that level for 4 h before decreasing to its is then used for computing the new values of meteoro- original value. logical parameters during the lifetime of the next feeder For the third feeder band, moist near-equatorial air band. parcels starting at ro 5 48 latitude reached the eyewall

Unauthenticated | Downloaded 09/29/21 04:37 AM UTC 32 MONTHLY WEATHER REVIEW VOLUME 138 at about 1700 UTC 21 March when the computed pc was 938 hPa. The Bureau’s estimate at that time was 947 hPa. Note that for the third feeder band also, near-equatorial air is computed to have moved in the CDO between 1400 and 1700 UTC 21 March in the same period of time that a sudden surface pressure fall outside the CDO was observed at Centre Island (see Fig. 13 and Table 2 of Part I). Note also that during that period, the intensification rate increased suddenly from 12 to 64 hPa day21. It stayed at that level for about 4 h before decreasing to 16 hPa day21. Results of computations during the lifetime of the fourth feeder band indicate that moist near-equatorial air starting at 48 latitude reached the edge of the CDO at 0330 UTC 22 March and the eyewall at 0630 UTC 22 March while the sudden surface pressure fall out- FIG. 5. Time variation of the computed mean gradient-level wind for selected radii for Tropical Cyclone Kathy. side the CDO occurred between 0500 and 0800 UTC 22 March. The calculated central surface pressure is 921.5 hPa while the Bureau’s estimate was 929 hPa d. The dissipation stage of Kathy (see Fig. 3 of Part I). The intensification rate also increased suddenly from 16 to 60 hPa day21 during this The intensification rate decreased gradually from time interval. 60 hPa day21 at 0600 UTC 22 March to 0 hPa day21 at Two series of computations were performed after 1200 UTC 22 March (see Fig. 13 of Part I). This sudden 0600 UTC 22 March using the angle of inflow that was decrease of intensification rate occurred at the time observed at the time of the analysis. One used a4 5 338 thecyclonehadjustlostmostofitssupplyofmoist and started from ro 5 2.08 latitude. The reason for that is near-equatorial air (see the second last paragraph of to calculate the central surface pressure and the sustained section 3a and Fig. 11q of Part I onward). Soon after maximum surface wind at about the time the cyclone was 1200 UTC 22 March Kathy started to weaken at a rate at its maximum intensity according to the Bureau’s es- of 30 hPa day21. This weakening process happened when timate. At 1130 UTC 22 March, the calculated central a low-level trough developed to the south of the cyclone surface pressure is 918 hPa, while the Bureau’s estimate (see Fig. 10 of Part I) and when a4, the angle of inflow for that time was 920 hPa. to the north of the cyclone had decreased to its pre- The other series of computations was performed with intensifying value of 208 (see Fig. 2), and also when rel- a4 5 208 and starting at ro 5 38 latitude to determine the atively dry trade air was entering the cyclone circulation. intensity of the cyclone when it passed over Centre Island. At 2000 UTC 22 March the cyclone center was about 5 km from the Centre Island meteorological 4. The radial wind profile station. The computed minimum central surface pres- a. The time variation of the radial surface wind profile sure for that time is 936 hPa, while the recorded minimum surface pressure at the meteorological station Marine engineers require a good forecast of the radial was 940 hPa (Murphy 1985). The computed mean distribution of surface winds around a tropical cyclone gradient-level wind at the radius of maximum wind is to forecast wave heights, particularly when the cyclone is 64.7 m s21. moving across a coastline. In the present model, gradient- To determine the sustained maximum mean surface level mean winds averaged around the cyclone, calcu- wind ahead of the cyclone, we use an asymmetry factor lated at different times for a number of selected radii [see of 0.83 (see Table S1 in the supplement) and obtain e.g., Table S5(b) in the supplement] have been used to a maximum wind of (0.83 3 64.7) or 53.7 m s21. The prepare Fig. 5 that shows the time variation of the radial peak gust is (1.25 3 53.7) or 67 m s21 (see section 2b for mean gradient-level wind profile. By multiplying the the gust factor of 1.25). The sustained maximum mean gradient-level wind by an asymmetry factor (Table S1 in surface wind recorded at Centre Island ahead of the the supplement), the radial distribution of the surface cyclone was 52 m s21 and the recorded peak gust was winds can be obtained for any orientation relative to the 64 m s21, quite close to the computed values. direction of motion of the cyclone.

Unauthenticated | Downloaded 09/29/21 04:37 AM UTC JANUARY 2010 L A J O I E A N D W A L S H 33 b. The radial surface wind profile ahead of Kathy Since the best track of Kathy (Fig. 2 of Part I) indicates that at and after 1500 UTC 22 March when Kathy was approaching and passing Centre Island it was moving at 19 km h21, a radial profile of sustained mean surface wind ahead of the cyclone can be determined. The distances of the cyclone center from the island were determined at half hour intervals between 1500 and 1930 UTC 22 March. The observed 10-min sustained mean surface wind for each of these distances ahead of the cyclone was obtained from the anemogram of Centre Island (not shown). The mean gradient-level wind (averaged around the cyclone) was estimated from the computed winds. The computed sustained mean surface wind ahead of the cyclone for each of these times was determined as follows. Between rm and 60 km the surface winds are ob- FIG. 6. Variation with distance of observed 10-min sustained tained by reducing the computed mean gradient-level mean surface winds, as read from the anemogram of Centre Island winds by the asymmetry adjustment factor of 0.83 (0.8 3 (solid line) and computed mean surface winds (dashed line) ahead 1.04, see section 2b) to obtain the sustained mean surface of Tropical Cyclone Kathy. winds ahead of the cyclone. For distances greater than 60 km an asymmetry adjustment factor of 0.73 (0.7 3 tral surface pressure of 935 hPa. The eye crossed the 1.04, see Table S1 in the supplement) is used. This is north Queensland coast near Innisfail between 2020 and because the factor k (to reduce gradient-level winds to 2120 UTC 19 March 2006. Satellite data from the U.S. the surface) is 0.7 instead of 0.8 at distances .5r m Naval Research Laboratory were used to estimate the (Georgiou 1985), and for winds far outside the eyewall time the cyclone started to develop, the size of r and r (Franklin et al. 2000, 2003). The comparison between e m following the technique of Lajoie and Walsh (2008). the computed and observed winds ahead of the cyclone Here r was estimated from TMI data and r was found is shown in Fig. 6. e m to be 33 km at first, but decreased to 20 km a few hours The times of the observed and computed winds, before the cyclone crossed the coast. Hourly radar pic- sometimes differ by about 1 h, but this is of no great tures indicate that the eye was shrinking gradually to consequence, since as can be verified in Fig. 5, the sus- 15 km when the cyclone crossed the coast. The angle of tained maximum wind changed only by about 1 m s21 inflow of the northerly airstream to the north and east- in 1 h. The computed winds were found to be within northeast of the cyclone was 308 for the first 30 h of in- 2ms21 of the observed winds up to a distance of 80 km tensification and 458 thereafter until the cyclone reached from the cyclone center. the coast. The initial radial mean profile of the gradient- level wind speed to the north and northeast of the de- 5. Testing the proposed model pression was estimated from the Darwin objective In the next two sections, the equations that we have tropical streamline and isotach analyses and is shown in developed for Kathy to calculate the central surface Table S7 (in the supplement), which also shows the pressure, maximum wind, the radial distribution of the computed surface pressure profile. sustained mean surface wind around the cyclone, and All the computations for the intensity of Tropical their changes with time, have been applied to Tropical Cyclone Larry were performed using the same equa- Cyclones Larry and Oliver with a view to verifying their tions that were developed to evaluate the intensity of general applicability. Tropical Cyclone Kathy, and can be perused in Lajoie (2007). The computed central surface pressure (dashed), a. Tropical Cyclone Larry and the estimated central surface pressure by the Bu- Larry started as a tropical depression in the eastern reau (dotted) and by the Joint Typhoon Warning Center Coral Sea. It started to intensify at about 0900 UTC (JTWC) (solid) are shown in Fig. 7. The JTWC esti- 17 March 2006 at about 1300 km east of Cairns. The mates were obtained in tropical cyclone warnings issued depression moved in a general westerly direction, while by the Marine Meteorological Division of the Naval intensifying slowly at first and more rapidly later to Research Laboratory. The computed central surface become a small but intense tropical cyclone with a cen- pressures were closer to the estimates of JTWC than

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tropical cyclone Larry for selected radii is shown in Fig. 8. The titles (b), (c), and (d) on the y axis on the right-hand side of Fig. 8 are, respectively, for the sustained mean surface wind ahead, to the left of the direction of motion (DOM), and for the maximum gust to the left of the DOM. This diagram therefore shows the time variation of the radial computed mean gradient-level wind proﬁle around Tropical Cyclone Larry. Maximum measured sustained mean winds and maximum measured wind gusts at 10 different locations are mentioned in the Queensland Regional ofﬁce tropical cyclone report (Bureau of Meteorology 2007). The wind-reporting stations are spread about 100 km on either side of the cyclone track. Of the 10 wind-reporting stations, 1 is on Flinders Reef (17.728S, 148.458E), 2 are

FIG. 7. Time variation of the central surface pressure n 18–20 Mar on little islands that are less than 3 m above mean sea 2006: computed values (dashed), estimates from JTWC (solid), and level, and the other 7 are inland stations along the coast. those from the Bureau (dotted). Table 1 gives the names of these wind-reporting stations, the time at which they experienced the maximum wind those of the Bureau, except when the cyclone was about or gust, their orientation with respect to the direction of to cross the coast, when the computed and the Bureau’s motion (R to the right, L to the left of the cyclone track central surface pressures were approximately the same. and A ahead of the cyclone), their distance from the The sustained maximum mean surface winds, esti- cyclone center at the time of the maximum winds, the mated by JTWC in their cyclone warnings at different computed mean gradient-level wind at that distance, times, were also very close to the computed ones, the the computed surface wind [using k 5 0.8 and Table S1 mean difference between the two being 4 kt. (in the supplement) to take into account the general The time variation from 18 to 20 March 2006 of the asymmetric wind distribution around a tropical cyclone], computed mean gradient-level wind averaged around the measured maximum surface wind, and the computed

FIG. 8. Time variation of (a) the computed mean gradient-level wind averaged around Tropical Cyclone Larry 18–19 Mar 2006. The diagram also shows for different times the radial proﬁle of (b) the sustained mean surface wind ahead of the cyclone, (c) the sustained mean surface wind to the left of the direction of motion (DOM), and (d) the maximum gust to the left of the DOM.

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TABLE 1. Computed and measured maximum sustained mean surface winds and maximum gusts at observing stations within 100 km of the center of Tropical Cyclone Larry when it crossed the north Queensland coast in Mar 2006.

Computed Computed Measured gradient max mean max mean Computed Measured Name of Distance level winds surface winds surface wind max gust max gust station Time and Date Orientation (8 lat) (m s21) (m s21) (m s21) (m s21) (m s21) Low Isles 2300 UTC 19 Mar R 1 29.57 21 20 26.3 21 Green Island 2104 UTC 19 Mar R 0.7 32.9 23.3 23.6 29.2 30.3 Flinders Reef 1110 UTC 19 Mar 47.6 43.3 43.1 54.1 58.6 Cairns Airport 2201 UTC 19 Mar R 0.63 36.2 25.7 32.1 29.7 Mareeba 2239 UTC 19 Mar R 0.6 38.5 27.3 14.9 34.2 31.3 South Johnstone 2033 UTC 19 Mar A 0.1 58.8 48.8 20.5 61 50.3 Cardwell 1923 UTC 19 Mar L 0.8 33.1 29.2 24.6 36.5 — Lucinda 1932 UTC 19 Mar L 1 28.9 26.3 24.1 32.9 30.3 Ravenshoe Wind 2240 UTC 19 Mar A 0.3 53 44 22.6 55 51.3 Farm Bellinden Tower 7 2118 UTC 19 Mar L 1450 m 65 81.3 81.6 and the measured maximum gust. For the three small local factors. Because the maximum gust at Bellinden islands, the computed maximum mean surface winds are Tower 7 was measured at a height of 1450 m, the mean almost identical as those observed, and the computed gradient level wind has been used to compute the maxi- maximum gusts were 1 and 4.5 m s21 less than those mum gust and that agrees well with the measured gust. measured. For six of the low-lying wind-reporting inland stations, b. Tropical Cyclone Oliver their measured maximum mean winds were all less than 1) DATA AND ANALYSIS the corresponding computed values by 5.5 m s21 on the average. Their maximum measured gusts were, Oliver was a tropical cyclone that evolved in the Coral however, closer to the computed gusts; the average Sea from 4 to 12 February 1993. It moved southeast at difference between the two being 2.5 m s21. The dif- ﬁrst, then due south along 1528E to pass over the Lihou ference between the computed and the measured winds Reef Automatic Weather Station (AWS) at 17.528S, for inland stations is due to the fact that the factor k over 152.08E (see the best-determined track of Oliver in land is appreciably ,0.8 and may also depend on various Fig. 9). The AWS recorded winds and surface pressure

FIG. 9. Best track of Tropical Cyclone Oliver (Courtesy of the Australian Bureau of Meteorology).

Unauthenticated | Downloaded 09/29/21 04:37 AM UTC 36 MONTHLY WEATHER REVIEW VOLUME 138 every hour. Radar observations on Willis Island (16.38S, 150.08E) were used by the Queensland Regional ofﬁce to determine the distances and bearings of the cyclone center from the AWS every hour. The initial state of the environment just prior to the intensiﬁcation stage of the depression was determined from 12-hourly European Centre for Medium-Range Weather Forecasts (ECMWF) wind vectors at every 2.58 latitude and longitude and surface pressure analysis at 1800 UTC 4 February 1993. According to radar observations (i) the center of Oliver’s eye passed at 15 km east of the AWS; (ii) the AWS experienced the passage of a partial eye between about 1030 and 1430 UTC

7 February; (iii) re was 18 km; (iv) the cyclone was moving at 5.4 km h21; and (v) the maximum wind occurred at about 0730 UTC so that r , the radius of m FIG. 10. Time variation of the central surface pressure of Trop- maximum wind, was 5.4 3 5 or about 27 km. ical Cyclone Oliver 5–9 Feb 1993. The computed values are the To determine the cyclone intensity from our analyti- dashed line and the Bureau’s best-track estimates are the solid line. cal model, we need two other parameters: the angles of inﬂow in the northerly airstream in the northeast sector of the cyclone and the time at which the cyclone started for an air parcel to move from one selected radius to the to intensify. The angles of inﬂow at a radius of 48 latitude next. All the detailed computations of Oliver’s intensity

(a4) were estimated from the direction of the ECMWF can be found in Lajoie (2007). wind vectors that were 2.58 latitude and longitude apart. 2) RESULTS These angles of inflow were determined from two to four vectors selected between 18 and 58 latitude distance The time variation of the central surface pressure of from, and between north and northeast of, the cyclone Tropical Cyclone Oliver from 5 to 9 February 1993 is center. The a4 was 208 at 1200 UTC 4 February, 358 shown in Fig. 10. The dashed curve shows the variation between 0000 UTC 5 February and 0600 UTC 7 Feb- of the computed pc and the solid curve that of the Bu- ruary, 258 at 1200 UTC 7 February and 0000 UTC reau’s best-track estimates of pc. There is a difference of 8 February, and 208 at and after 1200 UTC 8 February. up to 6 hPa between the two curves when the cyclone These observations indicate that the northerlies, that was relatively far from the AWS, elsewhere the two formed part of the giant swirl (see section 2d of Part I) curves are almost coincident. Figure 10 shows that the became established between 1200 UTC 4 February and central surface pressure was constant between 0000 UTC 0000 UTC 5 February and we have assumed that the 7 February and 0600 UTC 8 February. cyclone started to intensify at 1800 UTC 4 February. The time variations of the computed mean gradient- The initial wind speeds at 48 to 18 latitude radii in the level winds (averaged around the cyclone) for six selected northerlies were estimated from 1200 UTC 4 February radii are shown in Fig. 11. The horizontal profile of the ECMWF wind vectors. Unfortunately because of the mean gradient-level wind at any time can be obtained coarse grid resolution, it was only possible to evaluate from Fig. 11. The latter also shows that the computed the wind speeds at 48 and 28 latitude radii. The speeds for maximum mean gradient-level wind was constant be- the other selected radii were interpolated. The initial tween 0000 UTC 7 February and 0900 UTC 8 February, winds, the surface pressure gradients required to main- indicating that the cyclone was in a steady-state condition tain them, and the surface pressure for the selected radii between these times. are given in Table S8 (in the supplement). The initial The radial profile of computed sustained mean sur- surface pressure at 48 latitude to the north and northeast face winds up to about 100 km along the direction of of the cyclone center was 1004 hPa. This was assumed to motion of the cyclone is now compared with the profile remain constant throughout the life of the cyclone. of observed surface winds (Fig. 12). The computed sus- We have used the radial profile of a of Fig. 2 and all tained mean surface wind speeds are obtained from the the equations that have been developed to compute the mean gradient-level wind and an asymmetry factor to intensity of Kathy, to compute (a) the wind speeds at reduce the gradient-level wind to the surface and to take selected radii, (b) the surface pressure gradients re- into consideration the general asymmetry of the wind quired to maintain these winds, and (c) the time intervals speeds around a tropical cyclone (see Table 1 of Part I).

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FIG. 11. Time variation of computed gradient-level mean winds at different radii for Tropical Cyclone Oliver 6–8 Feb 1993.

Observed AWS winds are represented by the solid The results of the computations for Tropical Cyclone curve and the computed winds by the dashed curve. The Oliver therefore strongly suggest that the interpreta- radial wind profiles ahead (before the cyclone center tions of the observations gathered in Tropical Cyclone passed over the AWS) are on the left-hand side of Fig. 12 Kathy are correct, and that the physical principles that and the wind profiles at the rear of the cyclone (after the cyclone center had passed over the AWS) on the right- hand side. It is gratifying to note that most of the computed winds within 100 km from the cyclone center are within 63 kt of the measured winds, both before and after the passage of the cyclone center. Such accuracy in the diagnosis of the radial surface wind profile could greatly help engineers to improve forecasting of wave heights. The archived satellite cloud data for Tropical Cyclone Oliver is not of good enough quality to study in detail the cloud structure inside and outside the CDO as was done for Kathy. A coarse examination of the pictures however indicates (a) that the feeder bands were rather short and did not originate near the equator and (b) that they did extend southward to merge with the CDO. This was followed by a suppression of convective clouds outside the CDO, but there was no marked increase in intensification rate as was observed in Kathy. It was also noted in Oliver that when it started to FIG. 12. Radial distribution of sustained mean surface winds weaken (approximately 0900 UTC 8 February), there within 100 km of the cyclone center of Tropical Cyclone Oliver, see text for the derivation of this wind distribution. Computed winds were no feeder bands feeding the cyclone, as they moved are the dashed line and observed winds are solid. The left-hand side east away from the cyclone. At the same time the angle of the diagram shows the wind profile ahead of the cyclone and the of inflow decreased to 208, as was observed in Kathy. right-hand side at the rear of the cyclone.

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TABLE 2. Meteorological parameters associated with the change in intensification rate of the three tropical cyclones studied. The N in column 1 is a reference number. The name and computed date and time of the cyclone are entered in columns 2 and 3. Column 4 lists the 21 change of intensification (hPa day ). Column 5 shows the status of a4 (i.e., whether it has remained constant or has changed). Column 6 shows ro, the radius from which air parcels had left to arrive at the eyewall at the time given in column 3. Columns 7 and 8 give the status of re and rm (i.e., whether they have remained constant when the intensification occurred or whether they have decreased or increased).

Name of Computed time Computed change of 21 N tropical cyclone (UTC) and date ›pc/›t (hPa day ) Status of a4 (8) ro (8 lat) Status of re (8 lat) Status of rm (8 lat) 1 Kathy 1230 20 Mar 210 to 225 Constant 4 2 0130 21 Mar 212 to 2100 Constant 4 $0.9–0.8 0.25–0.16 3 1600 21 Mar 211 to 264 Constant 4 Constant Constant 4 0300 22 Mar 216 to 262 Constant 3.5 0.06–0.05 0.13–0.12 5 1400 22 Mar 0 to 130 42.5–20 2.5 Constant Constant 6 Larry 1500 18 Mar 224 to 236 Constant 4 $0.32–0.17 0.5–0.3 7 0930 19 Mar 236 to 264 30–45 4 Constant Constant 8 Oliver 0200 7 Feb 228 to 0 35–25 4 Constant Constant 9 1030 8 Feb 0 to 112 25–20 4 Constant Constant

have been used to formulate the mathematical model (iii) intensification rate increases when a4 is constant are reasonably sound. and re and rm decreases (N 5 2, 4, and 6); (iv) intensification rate decreases when a4 decreases (N 5 8) and the tropical cyclone starts to dissipate 6. Intensification rate a few hours after the decrease of a4 to its pre- We now examine the parameters that are associated intensifying value of 208 (N 5 5 and 9). with the rate of intensification, defined here as the rate at In Kathy and in Oliver (not shown), when a changed to which the central surface pressure changes with time. 4 208 the feeder bands moved away from the cyclones and The slope of the tangent at any point on the p time c stopped supplying them with moist near-equatorial air curve of Fig. 3 of Part I, and Figs. 7 and 10 of this paper is and the cyclones started to weaken. a measure of the rate of intensification of the cyclone at The foregoing paragraphs highlight the importance of that particular time. the import of moist near equatorial air into the core of Importance of the import of moist near-equatorial air a tropical cyclone on its intensification rate and strongly into the cyclone core on intensification rate suggest that it is necessary to monitor a4, re, and rm and their changes to diagnose or forecast the changes in in- With a view to studying whether changes in a4, re, and tensification rate. rm are associated with a change in the intensification rate of a tropical cyclone, we have prepared Table 2 from the 7. Discussion and conclusions computed data in Figs. 3 and 13 of Part I, and Figs. 7 The simple mathematical model that has been de- and 10 of this paper. Only significant changes in in- veloped following the sequence of observed features of tensification rate are included in Table 2. Figure 13 of severe Tropical Cyclone Kathy, and that has given sur- Part I shows how the intensification rate varies with time prisingly good results in diagnosing the time variation in intense Tropical Cyclone Kathy (19–22 March 1984). of its intensity, has been used to diagnose successfully As can be noted the intensification rate increased sud- the time variation of the intensity of Tropical Cyclones denly, then stayed constant for a few hours before de- Larry and Oliver. creasing gradually to its former value. At one stage the The model handled well the time variation of the intensification rate reached 100 hPa day21. The maxi- central surface pressure of the three tropical cyclones; mum intensification rate for Larry (18–19 March 2006) the computed surface pressures decreased when the and Oliver (5 February 1993) was appreciably less. cyclones were intensifying and increased when the cy- The facts that emerge from Table 2 are as follows: clones started to dissipate. The central surface pressures

(i) intensiﬁcation rate increases when ro is $3.58 lati- also corroborated very well with observed central surface tude (mostly 48 latitude; see N 5 1–4, 6–9). In all pressures when the cyclones passed close to an observing

25 cases when ro # 3.08 latitude, there was no sig- station. The computed maximum winds agreed very well niﬁcant change of intensiﬁcation; with observed ones. The radial distribution of the sus-

(ii) intensiﬁcation rate increases when a4 is constant or tained mean winds around the cyclones (after taking into increases (N 5 1–4, 6, and 7); account the general wind speed asymmetry around a

Unauthenticated | Downloaded 09/29/21 04:37 AM UTC JANUARY 2010 L A J O I E A N D W A L S H 39 tropical cyclone) was also practically the same as the mosphere, the degree of convective activity and the observed radial wind proﬁle, particularly for Kathy and amount of latent heat released, the vertical wind shear, Oliver for which we have ground truth data. the upper-troposphere trough interaction, and the vor-

It is instructive to compare our wind field model tex Rossby waves all act together to control the sizes of re methodology to that employed in the recent work of and rm and hence of pc. Also, a4, the angle of inflow of Knaff and Zehr (2007), Kossin et al. (2007), and Holland the northerlies at 48 latitude to the north and northeast (2008). Knaff and Zehr (2007) used statistical methods of the system, appears to be important for its development to find the best fit to observations for several predictors, and intensification: the greater the inflow angle, the including environmental pressure, storm motion, lati- greater the intensification rate. tude, storm size, and intensification trend. Kossin et al. In the North Atlantic region there are a great number (2007) estimates the two-dimensional wind profile of an of conventional data as well as data from many buoys individual storm from a principal component analysis of and reconnaissance flights around a hurricane that its IR image that is supplied as input, along with storm makes this region ideal for testing the efficaciousness of intensity and latitude, to a regression model derived from a technique. Unfortunately however, the circulation and previous observations. Holland (2008) specifies a shape cloud structure around a hurricane is not always the parameter that is as a function of central pressure drop, same as those in the Australian tropical cyclone region. rate of intensification, latitude, and speed of translation. In the latter region the extra source of appreciable Our method, in contrast, uses previously determined amount of moisture, apart from the ocean surface, is to values of the eye radius and the radius of maximum wind, the north of the cyclone. In the North Atlantic, however, along with estimates of the inflow angle and assumed hurricanes have two sources of heat and moisture: one to forms of the inflow trajectories. Thus, it uses quite dif- the south and the other to the north due to the presence ferent assumptions. Despite this, we show in Table S9 (in of the warm waters of the Gulf Stream. Thus, while the supplement) that the maximum winds computed from a tropical cyclone in the Australian region imports high- the model are close to those of Holland (2008) for the energy air only from the north via a single set of feeder cyclones examined here if we use our computed surface bands, the Atlantic hurricane is sometimes concurrently pressure drop, and both are quite similar to observations. being fed with high-energy air through two sets of feeder The success of correctly diagnosing the main intensity bands: one to the south and the other to the north. The parameters for the three tropical cyclones by the model model proposed here would therefore require some presented in this paper rests mainly in the physical in- modifications before being applied to North Atlantic terpretations of the changes that occurred in the large- hurricanes. It would be quite interesting, however, if the scale circulation and in the cloud structure of Tropical skill of this proposed model could be evaluated in other Cyclone Kathy during its intensification and dissipation tropical cyclone basins where there is only one source of stages. It is suggested that the main features that have high-energy air. been taken into account in the proposed model and that Another possible limitation of this model is caused by have contributed to the surprisingly good results are a4, the assumption that the surface pressure remains con- the angle of inflow at 48 latitude to the north and stant at a radius of 48 (see section 2a). Some large cy- northeast of the cyclone; the assumptions regarding the clones may have circulations that violate this assumption, forms of air parcel trajectories in the four defined re- thus requiring a correction to be made to the central gions of the storms; the assumed import by the feeder pressure calculation. We have not yet investigated the bands of moist air into the center of the storms, causing impact of this effect. cyclone intensification (Kelley and Stout 2004); and It is not known what causes the circulation change at a good estimate of re and rm that makes it possible to the start of intensification. But if a4, re, and rm and their calculate Vrm and pc. changes can be monitored, the proposed technique of- The correct forecasts of the maximum winds for the fers a method for diagnosing and forecasting with a lead three cyclones therefore indicate that the maximum time of at least 15 h (30 h if there is no change in the wind is partly dependent on the environmental condition environmental circulation after the first 15 h) the central and partly on rm. surface pressure, the maximum sustained mean surface The fact that the computed central surface pressures wind and maximum gust, and the radial surface wind were correctly diagnosed, being close to, if not exactly distribution around the cyclone. as, the corresponding observed values, suggests that all factors that influence the central surface pressure of Acknowledgments. We are indebted to the Director of a tropical cyclone, such as the sea surface temperature, the Australian Bureau of Meteorology for the use of the relative humidity in the low and midlevel of the at- some of the data used in this study, to Mr. Jeff Callaghan

Unauthenticated | Downloaded 09/29/21 04:37 AM UTC 40 MONTHLY WEATHER REVIEW VOLUME 138 of the Queensland Meteorological office for providing ——, 2008: A revised hurricane pressure-wind model. Mon. Wea. the data that have been used to analyze the intensity of Rev., 136, 3432–3445. Tropical Cyclones Larry and Oliver. We are also in- Hughes, L. A., 1952: On the low-level wind structure of tropical storms. J. Meteor., 9, 422–428. debted to the National Hurricane Center and Joint Ty- Iguchi, T., T. Kozu, R. Meneghini, J. Awaka, and K. Okamoto, phoon Warning Center of the United States for using 2000: Rainprofiling algorithm for the TRMM precipitation their tropical cyclone warnings and to NOAA and NRL radar. J. Appl. Meteor., 39, 2038–2052. for their various types of online satellite data. We thank Jorgensen, D. P., 1984: Meso-scale and convective-scale charac- the University of Melbourne for supporting this re- teristics of mature hurricanes. 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