The Formation of Concentric Vorticity Structures in...

2722 VOLUME 61J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

The Formation of Concentric Vorticity Structures in Typhoons

H.-C. KUO

Department of Atmospheric Sciences, National Taiwan University, Taipei, Taiwan

L.-Y. LIN

National Science and Technology Center for Disaster Reduction, Taipei, Taiwan

C.-P. CHANG AND R. T. WILLIAMS

Department of Meteorology, Naval Postgraduate School, Monterey, California

(Manuscript received 12 December 2003, in final form 3 May 2004)

ABSTRACT

An important issue in the formation of concentric eyewalls in a tropical cyclone is the development of asymmetric structure from asymmetric convection. It is proposed herein, with the aid of a nondivergent barotropicmodel, that concentric vorticity structures result from the interaction between a small and strong inner vortex(the tropical cyclone core) and neighboring weak vortices (the vorticity induced by the moist convection outsidethe central vortex of a tropical cyclone). The results highlight the pivotal role of the vorticity strength of theinner core vortex in maintaining itself, and in stretching, organizing, and stabilizing the outer vorticity field.Specifically, the core vortex induces a differential rotation across the large and weak vortex to strain out thelatter into a vorticity band surrounding the former. The straining out of a large, weak vortex into a concentricvorticity band can also result in the contraction of the outer tangential wind maximum. The stability of the outerband is related to the Fjørtoft sufficient condition for stability because the strong inner vortex can cause thewind at the inner edge to be stronger than the outer edge, which allows the vorticity band and therefore theconcentric structure to be sustained. Moreover, the inner vortex must possess high vorticity not only to bemaintained against any deformation field induced by the outer vortices but also to maintain a smaller enstrophycascade and to resist the merger process into a monopole. The negative vorticity anomaly in the moat servesas a ‘‘shield’’ or a barrier to the farther inward mixing the outer vorticity field. The binary vortex experimentsdescribed in this paper suggest that the formation of a concentric vorticity structure requires 1) a very strongcore vortex with a vorticity at least 6 times stronger than the neighboring vortices, 2) a large neighboring vorticityarea that is larger than the core vortex, and 3) a separation distance between the neighboring vorticity field andthe core vortex that is within 3 to 4 times the core vortex radius.

1. Introduction

Aircraft observations (e.g., Willoughby et al. 1982;Black and Willoughby 1992, hereafter BW92) show thatintense tropical cyclones often exhibit concentric eye-wall patterns in their radar reflectivity. In this patterndeep convection within the inner, or primary, eyewallis surrounded by a nearly echo-free moat, which in turnis surrounded by a partial or complete ring of deepconvection. Both convective regions typically containwell-defined local wind maxima. The primary windmaximum is associated with the inner core vortex, whilethe secondary wind maximum is usually associated withan enhanced vorticity field embedded in the outer rain-band. An example of the concentric eyewalls in Hur-

Corresponding author address: H.-C. Kuo, Department of At-mospheric Sciences, National Taiwan University, Taipei 106, Taiwan.E-mail: [email protected]

ricane Gilbert (1988) is given in detail by Willoughbyet al. (1989) and BW92. Approximately 12 h after reach-ing its minimum sea level pressure of 888 hPa, the low-est recorded so far in the Atlantic basin (Willoughby etal. 1989), Hurricane Gilbert displayed concentric eye-walls. BW92 estimated the radius to be between 8–20km for the inner eyewall and 55–100 km for the outereyewall. Between the two eyewalls, an echo-free gap(or moat) of about 35 km exists where the vorticity islow. Aircraft radial observations also showed the con-traction of the outer tangential wind maximum from adistance of 90 km from the storm center to 60 km inapproximately 12 h (e.g., Fig. 7 of BW92). Moreover,the core vortex intensity remained approximately thesame during the contraction of the outer tangential windmaximum.

Shapiro and Willoughby (1982) and Schubert andHack (1982) used a simple symmetric model of balanced

15 NOVEMBER 2004 2723K U O E T A L .

vortex response to specified heating to propose that heat-ing–vorticity interaction can lead to convective-ringcontraction. Their mechanisms involved both the spec-ified diabatic heating and the inertial stability structure.If the ring contains active convective heating, the mostrapid increase in windspeed lies on the inward side ofthe wind maximum and the ring may contract with time.However, the formation of a concentric eyewall wasoften observed to start from the organization of asym-metric convection outside the primary eyewall into aband that encircled the eyewalls (e.g., Fig. 3 of BW92).It is not yet clear how the symmetric models may beextended to explain the formation of concentric eyewallsfrom asymmetric convection. Montgomery and Kallen-bach (1997) proposed that the concentric eyewalls mightbe the result of radially propagating linear vortex Ross-by waves and the presence of a critical radius in thetropical cyclone. The radially varying vorticity assumesthe role of the meridional gradient of the Coriolis pa-rameter. Unlike the planetary Rossby waves that canpropagate over large meridional distances, the vortexRossby waves are more confined to the radius of max-imum winds in the tropical cyclone and therefore theirrole in the contraction of outer bands from a distanceof the order 100 km from the storm center may be lim-ited. Kossin et al. (2000) investigated the dynamic sta-bility of concentric vorticity structures in tropical cy-clones with a nondivergent barotropic model. Theirstudy shed light on the interactions between a tropicalcyclone’s primary eyewall and a secondary ring of en-hanced vorticity, but the question of the formation ofconcentric vorticity structures was not discussed. Re-cently, Nong and Emanuel (2003) have examined thedynamics of axisymmetric concentric eyewall cycles inthe context of axisymmetric models. Their study indi-cates that the secondary eyewalls may result from afinite-amplitude wind-induced surface heat exchange(WISHE) instability, triggered by external forcings.Asymmetric dynamics processes that are intrinsic to thehurricane vortex are not included in their axisymmetricmodel.

In this paper we show that the organization of theasymmetric convection into a symmetric concentric eye-wall can be accomplished through a binary vortex in-teraction between a small and strong inner vortex (thetropical cyclone core) and neighboring weak vortices(the vorticity induced by the moist convection outsidethe central vortex). Our model is an extension of thoseof Dritschel and Waugh (1992) and Dritschel (1995),who described the general interaction of two barotropicvortices with equal vorticity but different sizes. Theyconducted experiments on the f plane by varying theratio of the vortex radii and the distance between theedges of the vortices normalized by the radius of thelarger vortex. The resulting structures can be classifiedinto elastic interaction, merger, and straining-out re-gimes. In the complete straining-out regime, a thin re-gion of filamented vorticity bands surrounding the cen-

tral vortex with no incorporation into the central vortexappeared to resemble a concentric vorticity structure.However, the outer bands, which result from the smallervortex, are much too thin to be identified with that ob-served in the outer eyewall of a tropical cyclones. Inradar observations of Typhoon Lekima of 2001 (Fig.1), we noticed a huge area of convection outside thecore vortex that wraps around the inner eyewall to formthe concentric eyewalls in a time scale of 12 h. A similarexample may be found in Figs. 2–9 of Hoose and Colon(1970). In these cases the vorticity in the large areaoutside the core appear to be much weaker than that inthe small core area, a situation that was not included inDritschel and Waugh’s study.

In this study a nonlinear barotropic model is used toextend Dritschel and Waugh’s (1992) and Dritschel’s(1995) study by adding vorticity ratio as a third externalparameter, in addition to the radii ratio and the nor-malized distance between the two vortices. It will beshown that considering this difference of the vorticityof the two vortices is crucial in the formation of con-centric vorticity structures. Namely, one way to producea halo of enhanced vorticity around an intense vortexis through a binary interaction in which the large, weakvortex is completely strained out. It will be shown thatthis mode of interaction is most likely to occur whenthe peak vorticity in the small, strong vortex is at least6 times that of the large, weak vortex. Section 2 de-scribes the solution method and the model parameters.The numerical results are presented in section 3, andthe concluding remarks are given in section 4.

2. Model and initial conditions

The basic dynamics considered are two-dimensionalnondivergent barotropic with ordinary diffusion; that is,Dz/Dt 5 n¹2z, where D/Dt 5 ]/]t 1 u(]/]x) 1y(]/]y). Expressing the velocity components in terms ofthe streamfunction by u 5 2]c/]y and y 5 ]c/]x, wecan write the nondivergent barotropic model as

]z ](c, z )21 5 n¹ z, (1)

]t ](x, y)

where2z 5 ¹ c (2)

is the invertibility principle and ]( , )/](x, y) is the Ja-cobian operator. The diffusion term on the right-handside of (1) controls the spectral blocking associated withthe enstrophy cascade to higher wavenumbers. Similarto Prieto et al. (2001) and Kossin et al. (2000), we haveavoided the use of hyperviscosity [higher iterations ofthe Laplacian operator on the right-hand side of (1)]because of the unrealistic oscillations it can cause in thevorticity field. Three integral properties that we shallmonitor during our numerical simulations are the energyE 5 ## ½=c · =c dx dy, the enstrophy Z 5 ## ½z2 dxdy, and the palinstrophy P 5 ## ½=z · =z dx dy, a mea-


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FIG. 2. Initial schematic configuration of two circular vortices withradii R1 and R 2 (R1 , R 2 ), vorticity z1 and z 2 (z1 . z 2 ), and thegap D.

sure of the overall vorticity gradient in the domain. Ascan be shown from (1) and (2), these three integralproperties are related by

dE5 22nZ, (3)

dt

dZ5 22nP. (4)

dt

We perform calculations on the doubly periodic fplane. The discretization of the model is based on theFourier pseudospectral method, with 512 3 512 equallyspaced collocation points on a 200 km 3 200 km domainfor the binary vortex interaction experiments and a 600km 3 600 km domain for the Typhoon Lekima exper-iment. The code was run with a dealiased calculationof quadratic nonlinear terms with 170 3 170 Fouriermodes. Time differencing was via the fourth-orderRunge–Kutta method with a 6-s time step. The diffusioncoefficient, unless otherwise specified, was chosen tobe n 5 6.5 m2 s21. For the 200 km 3 200 km domainthis value of n gives an e21 damping time of 45 minfor all modes having total wavenumber 170, and adamping time of 3 h for modes having total wavenumber85. Some of the experiments were performed at doubledthe domain size. Results were found not to be sensitiveto the domain size.

We consider initial conditions consisting of J Ran-kine-like vortex patches,1 that is,

J

z(x, y, 0) 5 z P(r ), (5)O j jj51

where z j is the vorticity in the jth vortex patch; rj 5 [(x2 xj)2 1 (y 2 yj)2]1/2/Rj is a nondimensional radialcoordinate; xj, yj are the center coordinates; Rj is theradius of the jth vortex patch; and

30 11 2 exp 2 exp , if r , 1,j1 2 [ ]r r 2 1j jP(r ) 5j

0, otherwise,

(6)

is an analytical approximation to the unit step function,which has been introduced to reduce the Gibbs phe-

1 The Rankine vortices are with zero vorticity gradient and rapiddecrease of angular velocity with radius outside the core. DeMariaand Chan (1984) argued that mergers in binary vortex interaction canalso occur due to vortex propagation on the outer vorticity gradientsassociated with each vortex. The interaction of the tangential windfield with the outer vorticity field of the opposite vortex adds a com-ponent to the motion, which can cause the separation distance toeither decrease or increase, depending on the direction of the vorticitygradient. Thus, the extended vorticity gradient in a more realisticvortex should make merger more likely (or less likely) than with theRankine structure. Moreover, the slower decrease of angular velocityassociated with the extended vorticity gradient should slow the fil-amentation process.

nomenon in the initial condition. Our experiments in-clude both binary vortex interactions (i.e., J 5 2) andmultiple vortex interactions. For the large, weak vortexwe assume z2 5 3 3 1023 s21 and for the small, intensevortex we assume R1 5 10 km. For the binary vortexinteraction experiments with d as the distance betweenthe vortex centers, the binary vortices are specified bychoosing numerical values for the dimensionless gap

D d 2 (R 1 R )1 25 , (7)R R1 1

the vorticity strength ratio

z1g 5 , (8)z2

and the vortex radius ratio

R1r 5 . (9)R2

Figure 2 depicts the parameters for the binary vortexexperiments. The parameter ranges studied in this paperare 0 # D/R1 # 4, 1 # g # 10, and 1/4 # r # 1.

3. Numerical results

a. Binary vortex interaction

We consider the binary vortex interaction similar toDritschel and Waugh (1992) except that we have thevortex strength ratio as an additional control parameter.Specifically, we use a small and strong vortex to rep-resent the tropical cyclone core vortex and a large anda weaker vortex to represent the relatively weak vor-ticity induced by the moist convection outside the cen-tral vortex of a tropical cyclone. The idealization stemsfrom the Typhoon Lekima observation that the eye corewas surrounded by a huge area of convection before theformation of concentric eyewalls. Figure 3 shows thesensitivity of the vorticity field with respect to the vor-


FIG. 3. The sensitivity of the vorticity field in the binary vortexexperiments with respect to the vorticity strength ratio (g) at hours0, 3, 6, and 12, with the dimensionless gap D/R1 5 1, and the vortexradius ratio r 5 1/3.

ticity strength ratio (g) at hours 0, 3, 6, and 12 with thedimensionless gap D/R1 5 1, and the vortex radius ratior 5 1/3. It is clear from Fig. 3 that the two vorticesundergo a behavior ranging from merger (g 5 1 and g5 3), to tripole formation (g 5 5), to concentric vor-ticity structure with a moat (g 5 6). The tripole is anelliptical inner vortex and two distinctive minima in themoat. The last regime explains the results of Ritchie andHolland (1993) who included the interaction between asmall, strong vortex and a large, weak vortex in one oftheir experiments. They did not produce a concentricvorticity structure apparently because their vorticitystrength ratio did not exceed 3. Figure 3 suggests thatthe core vorticity of the small vortex is crucial in theformation of concentric vorticity structure. In the caseof concentric vorticity formation (g 5 6), we observea straining out of the weak vortex into a thin band thatspirals into and surrounds the strong core vortex at hour3, with subsequent development of concentric vorticitystructure and tightly wound spiral bands. This suggeststhat there are active merger dynamics occurring fromhour 3 until the more stable and coherent concentricvorticity structure is reached at hour 12. Kossin et al.(2000) investigated the dynamical stability of concentricvorticity structures in tropical cyclones with a nondiv-ergent barotropic model. Two types of instabilities wereidentified: 1) instability across the outer ring of en-hanced vorticity, and 2) instability across the moat. Type1 instability occurs when the outer vorticity band issufficiently narrow and the inner vortex is sufficientlyweak that it does not induce enough differential rotation

across the outer vorticity to stabilize it. Type 2 instabilityoccurs when the radial extent of the moat is narrow sothat barotropic instability may result. In the case of thetype 2 instability, Kossin et al. (2000) found that themoat and vortex evolve into a nearly steady tripole struc-ture. The formation of the tripole vortex in the g 5 5case apparently involves the straining out of the larger,weaker vortex into a finite width band surrounding thesmaller, stronger vortex, with a subsequent type 2 in-stability of wavenumber 2 across the moat. Two ex-amples of elliptical eyes that resemble the tripole vortexstructure were recently reported by Kuo et al. (1999)for the case of Typhon Herb (1996), and by Reasor etal. (2000) for the case of Hurricane Olivia (1994). Onthe other hand, neither type 1 nor type 2 instabilities(Kossin et al. 2000) are favored for g 5 6 and D/R1 51. Even though the change of sign of vorticity gradientacross the outer band satisfies the Rayleigh necessarycondition for barotropic stability, the band is stabilizedby the Fjørtoft sufficient condition for stability. Namely,the strong inner vortex causes the wind to be strongerat the inner edge than the outer edge, allowing the vor-ticity band and therefore the concentric structure to besustained. A similar mechanism is discussed by Drit-schel (1989) and Polvani and Plumb (1992), whoshowed how thin filaments can be stabilized by the flowfield of the main vortex. They argued that the filamentis linearly stable and appears circular in the presence ofsufficiently strong ‘‘adverse shear.’’ The adverse shearis an externally controlled parameter with the oppositesense as that produced by the filament’s vorticity alone.

Figure 4 is similar to Fig. 3 and shows the sensitivityof the vorticity field to the dimensionless gap D/R1 withthe vortex size ratio r 5 1/3 and the vorticity ratio g5 5. Figure 4 suggests that a moderate dimensionlessgap (e.g., D/R1 5 2) is favored for the formation of aconcentric vorticity structure. A weaker vortex that istoo far away leads to an elastic interaction, while aweaker vortex that is too close will lead to the formationof a tripole vortex. Complete merger occurs when g 55 and the gap vanishes. The simulation is in generalagreement with the stability analysis by Kossin et al.(2000). In their Fig. A1, the wavenumber 2 instability(which leads to the formation of a tripole vortex) has asharp boundary for the r1/r2 parameter, such that theinstability vanishes at r1/r2 # 0.55. The correspondingparameter of their r1/r2 is related to our dimensionlessgap D/R1 5 1/(r1/r2) 2 1. Thus, a larger dimensionlessgap of 2 (and thus a larger resultant moat size) will nothave type 2 instability, so that the concentric vorticitystructure can be maintained.

Figure 5 is similar to Fig. 3 and shows the sensitivityof the vorticity field to the vortex size ratio r with thedimensionless gap D/R1 5 1, and the vorticity ratio g 55. The concentric vorticity structure forms at r 5 1/2 andtripole vortices form with size ratios of 1/3 and 1/4.

Figure 6 is similar to Fig. 5 except that D/R1 5 0 andg 5 10. For all but the r 5 1 case, a concentric vorticity


FIG. 4. The sensitivity of the vorticity field in the binary vortexexperiments with respect to the dimensionless gap D/R1 at hours 0,3, 6, and 12 with the vorticity strength ratio (g 5 5) and the vortexradius ratio r 5 1/3.

FIG. 5. The sensitivity of the vorticity field in the binary vortexexperiments with respect to the vortex radius ratio r at hour 0, 3, 6,and 12 with the vorticity strength ratio (g 5 5) and the dimensionlessgap D/R1 5 1.

FIG. 6. Similar to Fig. 5, except that the dimensionless gap D/R1 50 and the vorticity strength ratio g 5 10.

structure forms when the core vortex is 10 times stron-ger. Other tests with g 5 10 and D/R1 up to 3 or 4 alsoyield concentric vorticity structures in every case inwhich the r parameter is smaller than unity (the weakervortex is larger in size than the core vortex.) The for-mation of the moat region at D/R1 5 0 and g 5 10occurs through the advection of the negative vorticityanomaly from the background vortex-free-region. Thestrong differential rotation outside the radius of maxi-mum wind of the core vortex may also contribute to theformation and maintenance of the moat. Rozoff et al.(2003, manuscript submitted to J. Atmos. Sci., hereafterR03) have examined the rapid filamentation zones inintense tropical cyclones. They argued that the strain-dominated flow region outside the radius of maximumwind of the core vortex can contribute significantly tothe moat dynamics.

Figure 7 shows the sensitivity of the vorticity fieldin the binary vortex experiments with respect to thediffusivity n at hours 0, 6, 12, and 36 when g 5 5, r5 1/3, and D/R1 5 2.5. Figure 7 suggests that concentricvorticity structures change to tripole structures at hour12 if we employ a diffusion that is 10 to 30 times largerthan n 5 6.5 m2 s21. The tripole structure in the highestdiffusion case (n 5 97.5 m2 s21) becomes a monopolestructure at hour 36, while the n 5 32.5 m2 s21 caseretains a tripole structure. The relatively small vorticityin the moat prohibits radial movement due to the dy-namics of inertial stability. The negative vorticity anom-aly in the moat serves as a ‘‘shield’’ to impose a barrierto the inward mixing of the outer vorticity field. On the

other hand, tripole structures may result from the re-duction of the moat size due to a larger n in the periodbetween hours 6 and 12. The wavenumber 2 growthfrom type 2 instability, as analyzed by Kossin et al.(2000), then sets the stage for tripole formation.

The time dependence of kinetic energy, enstrophy,


FIG. 7. The sensitivity of the vorticity field in the binary vortexexperiments with respect to the diffusivity n at hours 0, 6, 12, and36 with the vorticity strength ratio (g 5 5), vortex radius ratio r 51/3, and the dimensionless gap D/R1 5 2.5.

FIG. 8. Time dependence of (a) kinetic energy and enstrophy, and(b) palinstrophy for experiments in Fig. 7.

and palinstrophy for the experiments in Fig. 7 are shownin Fig. 8. The near conservation of the kinetic energy,the damping of the enstrophy field, and the initial in-crease and the eventual decrease of the palinstrophy fieldall possess the characteristics of two-dimensional tur-bulence. The larger values of n leading to the tripoleand monopole cases are associated with a more activeenstrophy cascade, as seen in Fig. 8. The small humpsin the palinstrophy around hour 34 for the n 5 6.5m2 s21 and n 5 3.25 m2 s21 cases are due to the strainingout of a small satellite vortex. The formation of a tripoleinstead of a concentric vortex, and the formation of amonopole with the largest n, appears to be related toselective decay of enstrophy versus kinetic energy andthe resulting merger process in two-dimensional tur-bulence (Batchelor 1969). In the case of a nearly in-viscid fluid (where n is very small), the vorticity con-tours can pack close together before diffusion is effec-tive. The closely packed contours increase | =z | , andhence the palinstrophy as shown in Fig. 8. Even whenn is small, the 22nP term on the right-hand side of (4)may not be small due to the increase of palinstrophy.We then have a significant enstrophy cascade. With sig-nificant enstrophy cascade (thus a smaller enstrophy lat-er), the right-hand side of kinetic energy equation (3)is small and the kinetic energy is nearly conserved.This is the phenomenon of the selective decay; that is,the enstrophy is selectively decayed over kinetic energy,in the two-dimensional turbulence (Cushman-Roisin

1994). In the presence of strong rotation, the wind fieldis nearly geostrophic, so that

Dpu ; , (10)

l

the kinetic energy is2Dp

2E ; u ; , (11)2l

and the enstrophy is

2 2u DpZ ; ; , (12)

41 2l l

where Dp is the pressure perturbation and l is the vortexscale. The near conservation of energy, as illustrated inFig. 8, according to (11), requires that Dp/l remainsapproximately constant. The cascade of enstrophy ac-cording to (12), along with the conservation of kineticenergy, imply a steady increase of l, with a proportionalincrease in Dp. Thus, the vortices become, on the av-erage, larger, stronger, and fewer. There is thus a naturaltendency toward larger structures with successive eddymergers. With every merger, energy is consolidated intolarger structures with concomitant enstrophy losses.Thus, the merger processes and the formation of mono-


FIG. 9. The sensitivity of the vorticity field in the binary vortexexperiments with the core vortices possessing the same maximumwind but different radius and vorticity field. Two vortices consideredhave the vorticity and radius of (1.5 3 1022 s21, 10 km) and (0.753 1022 s21, 20 km) respectively. The dimensionless gap is 1 in theexperiments. The outer vortices considered have the radius of 30 and40 km, respectively.

pole in a nearly inviscid fluid can be more significantwith a larger n. To conserve angular momentum and/orkinetic energy during the merger process, the inwardmerger of the vorticity field toward the core vortex mustbe accompanied also by some outward vorticity redis-tribution (Schubert et al. 1999). The outward redistri-bution can be seen in the form of filaments that orbitthe core vortex. On the other hand, coherent vorticitystructures such as the concentric vortex and the tripolevortex can prolong the merger process. The merger pro-cesses can be seen in our model results in which no-ticeable spiral bands exist at hours 3 and 6, but not athour 12 when the concentric vorticity and tripole struc-tures are formed. The results suggest that there are activemerger dynamics occurring at hours 3 and 6, after whicha more stable and coherent concentric vorticity structureis reached at hour 12. We observed in Fig. 7 that theconcentric vortex patterns change to tripole patternswhen n is increased. The increase of n, the increase ofenstrophy cascade or the increase of the merger processpresumably can reduce the moat size and lead to thewavenumber 2 instability. The increase of n can occurwhen tropical cyclones make landfall and the frictionfrom the boundary layer increases. Willoughby (1990)pointed out that the outer eyewall may not survive ifthe storm is close to land. The collapse of the concentriceyewalls, however, may not be adequately modeled withonly advective dynamics.

Figure 9 shows the results of experiments with thesame maximum wind in the core vortices but with var-iations in the core vortex radius and the maximum vor-ticity. Specifically, we considered the core vortices thatpossess the vorticity and radius of (1.8 3 1022 s21, 10km) and (0.9 3 1022 s21, 20 km), respectively. The pairof core vortices considered induce the same deformationfield or differential rotation in the region outside theradius of maximum wind. The dimensionless gap is 1in the experiments. The outer vortices considered havethe radius of 30 and 40 km, respectively. The g and rparameters in the first two experiments are (6, 1/3) and(3, 2/3), respectively. They indicate that a similar con-centric vorticity structure formed except that the double-size core vortex case possesses a thinner outer band.The thinner band is a result of more active merger ofouter vorticity into the core as well as a larger perimetersurrounding the core. The g and r parameters in thebottom half of Fig. 9 are (6, 1/4) and (3, 2/4), respec-tively. We observe that the smaller stronger vortex sim-ulation results in a tripole, while the corresponding caseresults in a monopole. In these experiments with thesame maximum tangential wind, the larger, weaker (interms of vorticity) inner vortices undergo more distor-tion and more active merger with the neighboring vor-tices. The experiments, along with the experimentsshown in Figs. 3, 5, and 6, suggest that the inner vortexhas to be strong not only to maintain itself against anydeformation field due to outer vortices, but also to pos-sess smaller enstrophy cascade and to resist the merger

process into a monopole. The resistance of the strongerinner vortex to deformation and merger agrees with 2Dturbulence experiments (McWilliams 1984) and with at-mospheric observations and theories (Bowman 1996;McIntyre 1989) that high vorticity gradients protect vor-tex cores from violent interaction. When the vorticitygradient is sharp, any radial flow will quickly producea large anomaly and quickly propagate away by thevortex Rossby wave along the edge before any furtherpenetration has occurred. An equivalent explanation isthat a high vorticity core exerts a high inertial stabilityand prevents radial penetration of the outside fluid intothe core. Figure 9 also supports the notion that the vor-ticity strength ratio, not the maximum tangential windratio, is the more useful experimental parameter.

Figure 10 gives the interaction regimes for binary vor-tices calculated as a function of the dimensionless gapD/R1 and the vorticity strength ratio z1/z2 for the radiusratios R1/R2 5 1/2, R1/R2 5 1/3, and R1/R2 5 1/4. Wehave classified the resulting interactions using the schemedevised by Dritschel and Waugh (1992). The structuresare categorized into the ‘‘concentric,’’ ‘‘tripole,’’ ‘‘merg-er,’’ and ‘‘elastic interaction’’ regimes. All calculationswere performed on the f plane. The abscissa in the two-dimensional parameter space in Fig. 10 is the dimen-sionless gap D/R1, which ranges from 0 to 4, and theordinate is the vorticity strength ratio g, which ranges


FIG. 10. Summary of numerical experiments with the parametersof the vorticity strength ratio (g ), the dimensionless gap D/R1, andthe vortex radius ratio r. The structures are categorized into the C(concentric), T (tripole), M (complete or partial merger), and EI (elas-tic interaction) regimes.

from 1 to 10. Figure 10 suggests that the demarcationzone between the concentric vorticity regime and themerger type regime is around g 5 5. The tripole vortexstructure is a distinct feature in the demarcation zone.Concentric vorticity structures are favored when g isgreater than 5. With g 5 6, concentric vorticity structuresoccur when the dimensionless gap (D/R1) ranges from0.5 to 3.5. The range of the dimensionless gap (D/R1) forthe concentric vorticity structure extends from 0 to 3.5for a g larger than 7. Of particular interest in the diagramis that when g is larger than 8, formation of a concentricvorticity structure requires a separation distance betweenthe neighboring vorticity field and the core vortex thatis within 3 to 4 times the core vortex radius.

Our simulations suggest that concentric vorticitystructures can be a result of binary vortex interaction.The flow fields associated with the strong core vortexprovide the necessary stretching, which can shear outthe weaker vortex into a thin strip of enhanced vorticitywrapped around the core vortex. The formation of con-centric vorticity structures requires a very strong corevortex with a vorticity at least 6 times stronger than theneighboring vortices, a large neighboring vorticity areathat is larger than the core vortex, and a separationdistance between the neighboring vorticity field and the

core vortex that is within 3 to 4 times the core vortexradius. If the separation distance is too small and thecore vortex is of only marginal strength, the resultantevolution most likely leads to monopole formation. Tri-pole vortex formation may result in the case of marginalstrength of the core vortex and a relatively small di-mensionless gap. When the vortex strength is greaterthan 8, a concentric vorticity structure can form evenwhen the dimensionless gap is zero initially.

b. Multiple vortex interaction and Lekimaexperiments

We have demonstrated with the binary vortex inter-action that concentric vorticity structures can form froma strong, small vortex and a weak, large vortex nearby.The strong, small vortex serves as the ‘‘organizer’’ ofthe surrounding vortices (the ‘‘satellites’’) into the con-centric vorticity structure. The purpose of studying mul-tiple vortex interactions is to investigate if there is apreferred surrounding vortex size (or preferred back-ground vorticity spatial scale) for the formation of theconcentric vorticity structure when the core vortex is ofsufficient strength. Specifically, we consider the sametotal vorticity but in different sizes in the satellites. Thetop row in Fig. 11 gives the benchmark binary vortexinteraction for comparison with the multiple vortex in-teractions. The benchmark binary vortex interaction hasthe vorticity strength ratio g 5 10, the vortex radiusratio r 5 1/4, and the dimensionless gap D/R1 5 1.0.The binary vortex interaction produces a concentric vor-ticity structure in the 12-h simulation. We then use thesame core vortex but split the surrounding vortex intotwo, four, and eight equally sized satellite vortices. Thetotal vorticity in these satellites is the same as the vor-ticity in the benchmark satellite. Figure 11 shows thatconcentric vorticity structures form in all of the multiplevortex interactions with two, four, and eight satellites.Other experiments with different patterns of the satel-lites (not shown) also yielded similar concentric vortic-ity structures.

Figure 12 is similar to the multiple vortex experimentsin Fig. 11 except the core is now surrounded by 9 and16 equally sized satellite vortices. The first two exper-iments are for the nine-symmetric-satellite orientationswith respect to the core organizer. The simulation in thefirst row in Fig. 12 suggests the formation of a con-centric vorticity structure. The second row in Fig. 12,with nine symmetric satellites farther away from thecore (D/R1 5 4.5), shows insufficient straining out athour 3 as compared to the first case. The subsequentevolution of the second row does not favor the formationof a concentric vorticity structure. The nine-satellite ex-periments suggest that even a symmetric initial satellitedistribution does not guarantee a concentric vorticitystructure if the separation distance is too large (e.g.,D/R1 5 4.5). In the case where the symmetric satellitesare too far away, there will be a weaker straining out


FIG. 11. The top row is the benchmark for the multiple vortexexperiments from the binary vortex experiment with the vorticitystrength ratio (g 5 10), vortex radius ratio r 5 1/4, and the dimen-sionless gap D/R1 5 1.0. The bottom rows are the multiple vortexinteractions with the same core vortex as the benchmark binary vortexinteraction. The total vorticity in the neighboring vortices is the sameas the neighboring vortex in the binary vortex experiment.

FIG. 12. The multiple vortex experiments with 9 and 16 neighboringvortices. The total neighboring vorticity is the same as in the bench-mark binary vortex experiment.

effect on the satellite vortices by the core organizer, aswell as insufficient stabilization by the core organizervortex. The third and fourth rows in Fig. 12 are for a16-satellite initial condition. Note that the surroundingvortices in the 16-satellite cases have the same radiusas the core vortex. The symmetric 16-satellite vortexexperiment (the third row) involves an initial conditionwhere the centers of the 16 satellites were on two con-centric circles (six on the inner circle and nine on theouter circle). The figure shows that a concentric vorticitystructure formed on the inner circle while the vorticeson the outer circle failed to form a secondary outer band.The asymmetric 16-satellite case (the fourth row) failedto produce a concentric vorticity structure. We have alsotested the 16-satellite initial condition in other asym-metric configurations and also with a very small dif-fusion (not shown) and found no concentric vorticitystructures. These results, in agreement with our binaryexperiments, support the notion that no concentric vor-ticity structure forms with satellite vortices of the sameradius as the core. The only exception to this is whenthe initial 16 satellites are symmetric and in a circleclose to the core vortex.

A summary of the multiple vortex experiments in-

dicates that there is a preferred threshold scale of thesurrounding vorticity for the formation of the concentricvorticity structure. The surrounding vorticity patchesshould be larger than the core vortex. Figure 13 showsthe tangential wind speed for radial arms toward thewest and south that emanate from the vortex center atdifferent times for the experiment in the second row ofFig. 11. The wind profiles show clearly a secondarymaxima in the tangential wind field contracting withtime in these different radial arms. Figure 13 also sug-gests the asymmetric nature of the contraction, as theinitial satellite vortices are located only on the west andsouth arms of the organizer vortex. The time and spatialscales of the secondary wind maximum contraction inFig. 13 are in general agreement with the observationsin Hurricane Gilbert (BW92). The contraction mecha-nism for the outer bands is often argued to be a balancedresponse to an axisymmetric ring of convective heating(Shapiro and Willoughby 1982; Schubert and Hack1982). Our results in Fig. 13 suggest that the nonlinearadvective dynamics involved in the straining out of alarge, weak vortex into a concentric vorticity band canalso result in the contraction of the secondary windmaximum.

Our final experiments involve a core organizer vortexand the surrounding vorticity field that resembles theshape of the convection observed in the Typhoon Lek-ima radar picture. Figure 14 shows sensitivity experi-


FIG. 13. The tangential wind speed for radial arms toward the west(left portion) and the south (right portion) that emanate from thevortex center at various times for the experiment in the second rowof Fig. 11.

FIG. 14. The Typhoon Lekima experiments with a core vortex andan area of weak vorticity that resembles of the shape of convectionas observed in the radar picture. The control experiment on the toprow has the vorticity strength ratio g 5 7.5, and the shortest distancefrom the outer vorticity boundary to the core vortex boundary isD/R1 5 0.6 (R1 5 10 km). The parameters in the second row are thesame as the control except with the parameter D/R1 5 0. The param-eters in the third row are the same as the control except with g 53. The parameters in the bottom row are D/R1 5 2.8 and g 5 3.

ments for the vorticity field at hours 0, 3, 6, and 12 forthese experiments. The control experiment on the toprow in Fig. 14 has the vorticity strength ratio g 5 7.5and the shortest distance from the outer vorticity bound-ary to the core vortex boundary is D/R1 5 0.6 (R1 510 km). The value g 5 7.5 is estimated from the radarradial wind and core vortex and with the assumptionthat the typical vorticity induced by convection is about10 times the local Coriolis parameter. The control ex-periment produces a concentric vorticity structure with-in 12 h. The second row in Fig. 14 is a similar exper-iment but with the parameter D/R1 5 0. In this exper-iment we find that a monopole results, possibly due toinsufficient negative vorticity being advected inward to

provide a shield from the inward mixing of the outervorticity field. The third row of Fig. 14 is the same asthe top row except that g 5 3. A monopole vortex isformed instead of a concentric vorticity structure in thisexperiment. The result is in agreement with the binaryvortex interaction in the sense that the core strength ismost vital not only to serve as an organizer but also toresist the merger process into a monopole. The last rowof Fig. 14 shows the results of an experiment withD/R1 5 2.8 and g 5 3. In this case the core vortex atthe center of a circle is approximately surrounded bythe inner boundary of the outer vorticity field. Consis-tent with the binary vortex interaction, the weak corevortex cannot maintain a concentric vorticity structureand it eventually becomes a monopole vortex.

4. Concluding remarks

There are many documented cases of binary tropicalcyclone interactions that resemble the theoretical workof Dritschel and Waugh (1992; e.g., see Larson 1975;Lander and Holland 1993; Kuo et al. 2000; Prieto et al.2003). The complete straining out regime of the binaryvortex interaction in Dritschel and Waugh (1992) showsa small, weaker vortex being sheared out into thin fil-aments of vorticity surrounding the large, stronger vor-


tex with no incorporation into the large vortex. Theregime resembles the concentric vorticity structure ex-cept the filaments are too thin to be called a concentriceyewall. Furthermore, Typhoon Lekima observationsindicate that it is a huge area of convection with weakcyclonic vorticity outside the core vortex that wrapsaround the inner eyewall, rather than the other wayaround. This large vortex forms the outer band on atime scale of 12 h. The interaction of a small and strongvortex with a large and weak vortex was not studied byDritschel and Waugh (1992) as their vortices are of thesame strength and their larger vortex was always the‘‘victor’’ and the smaller vortex was the one often beingpartially or totally destroyed. An extension of the com-plete straining out regime to include a finite-width outerband is needed to explain the interaction of a small andstrong vortex (representing the tropical cyclone core)with a large and weaker vortex (representing the vor-ticity induced by the moist convection outside the cen-tral vortex of a tropical cyclone). With the introductionof a parameter of vorticity strength ratio into the binaryvortex interaction problem, we have added a new di-mension to the Dritschel–Waugh vortex interactionscheme that provides a proper concentric vorticity struc-ture as well as the tripole vortex structure.

The vorticity strength of the central core vortex isessential in the formation of a concentric vorticity struc-ture. It has to be at least 6 times stronger than the neigh-boring vortices. In addition, the neighboring vorticityarea must be larger than the core vortex and with aseparation distance from the core vortex that is within3 to 4 times the core vortex radius. The contraction ofthe secondary tangential wind maximum and the for-mation of the moat are salient features of the binaryvortex interaction. The contraction of the secondary tan-gential wind maximum is in general agreement with theaircraft observations that show the contraction of theouter tangential wind maximum from a distance of 100to 50 km on a time scale of approximately 12 h (BW92).No diabatic heating is required in the present contractionmechanism. Diabatic heating, however, may be crucialin the enhancement of the secondary tangential windmaximum during the symmetrization of the asymmetricconvection. The negative vorticity anomaly in the moatserves as a barrier to the farther inward mixing of theouter vorticity field. The moat in our model is causedby the strong differential rotation associated with thecore vortex and the advective organization of the neg-ative vorticity anomalies. In nature, the strong subsi-dence induced by the intensified eyewall convectionmay also contribute to the formation of the moat (Dodgeet al. 1999). R03 hypothesized that both subsidence andrapid filamentation are important to the dynamics of themoat. All these arguments agree with the fact that theconcentric eyewalls often form when the tropical cy-clone is of sufficient strength.

The concentric vorticity structures in our barotropicnumerical experiments are quite robust and can maintain

themselves for more than 24 h after formation. No innervortex replacement cycle is modeled in our initial valueproblems. Presumably, a high-resolution ‘‘full-physicsmodel’’ that takes into account the moisture cutoff pro-cess is required to simulate the eyewall replacementprocess. With simple model calculations, our intent isnot to deprecate the importance of the moist physics,but rather to isolate the fundamental dynamics that maybe responsible for the concentric vorticity structure for-mation. Our arguments may be applicable to the for-mation of concentric eyewalls if the time scale of thecore vortex intensity modification is longer than the timescale of the concentric vorticity formation. Without de-tailed observations of the potential vorticity distributionin the eyewall evolution to compare with, our resultsmay remain suggestive. However, it does not seem un-reasonable to expect the pivotal role of the strength ofthe inner core vortex in maintaining itself, in stretching,organizing, and stabilizing the outer vorticity field, andthe shielding effect of the moat to prevent further mergerand enstrophy cascade processes in concentric eyewalldynamics.

Acknowledgments. This research was supported bythe National Research Council of Taiwan NSC91-2111-M-002-020 and NSC92-2625-Z-002-002 to NationalTaiwan University, and National Science Foundation,under Grant ATM 0101135 to the Naval PostgraduateSchool. We would like to thank Wayne Schubert, HughWilloughby, and an anonymous reviewer for their usefulcomments and suggestions.

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