VOL. 57, NO.24 JOURNAL OF THE ATMOSPHERIC SCIENCES 15 DECEMBER 2000
Unstable Interactions between a Hurricane's Primary Eyewall and a Secondary Ring of Enhanced Vorticity
JAMES P. K OSSIN,WAYNE H. SCHUBERT, AND MICHAEL T. M ONTGOMERY Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado
(Manuscript received 17 August 1999, in ®nal form 10 March 2000)
ABSTRACT Intense tropical cyclones often exhibit concentric eyewall patterns in their radar re¯ectivity. Deep convection within the inner, or primary, eyewall is surrounded by a nearly echo-free moat, which in turn is surrounded by an outer, or secondary ring of deep convection. Both convective regions typically contain well-de®ned tangential wind maxima. The primary wind maximum is associated with large vorticity just inside the radius of maximum wind, while the secondary wind maximum is usually associated with relatively enhanced vorticity embedded in the outer ring. In contrast, the moat is a region of low vorticity. If the vorticity pro®le across the eye and inner eyewall is approximated as monotonic, the resulting radial pro®le of vorticity still satis®es the Rayleigh necessary condition for instability as the radial gradient twice changes sign. Here the authors investigate the stability of such structures and, in the case of instability, simulate the nonlinear evolution into a more stable structure using a nondivergent barotropic model. Because the radial gradient of vorticity changes sign twice, two types of instability and vorticity rearrangement are identi®ed: 1) instability across the outer ring of enhanced vorticity, and 2) instability across the moat. Type 1 instability occurs when the outer ring of enhanced vorticity is suf®ciently narrow and when the circulation of the central vortex is suf®ciently weak (compared to the outer ring) that it does not induce enough differential rotation across the outer ring to stabilize it. The nonlinear mixing associated with type 1 instability results in a broader and weaker vorticity ring but still maintains a signi®cant secondary wind maximum. The central vortex induces strong differential rotation (and associated enstrophy cascade) in the moat region, which then acts as a barrier to inward mixing of small (but ®nite) amplitude asymmetric vorticity disturbances. Type 2 instability occurs when the radial extent of the moat is suf®ciently narrow so that unstable interactions may occur between the central vortex and the inner edge of the ring. Because the vortex-induced differential rotation across the ring is large when the ring is close to the vortex, type 2 instability typically precludes type 1 instability except in the case of very thin rings. The nonlinear mixing from type 2 instability perturbs the vortex into a variety of shapes. In the case of contracting rings of enhanced vorticity, the vortex and moat typically evolve into a nearly steady tripole structure, thereby offering a mechanism for the formation and persistence of elliptical eyewalls.
1. Introduction were as shown in Fig. 1. At this time the storm had concentric eyewalls. The inner eyewall was between 8- During the period 11±17 September 1988, Hurricane and 20-km radius and the outer eyewall between 55- Gilbert moved westward across the Caribbean Sea, over and 100-km radius, with a 35-km echo-free gap (or the tip of the Yucatan peninsula, and across the Gulf of moat) between the inner and outer eyewalls. The inner Mexico, making landfall just south of Brownsville, Tex- tangential wind maximum was 66±69 m sϪ1 at 10-km as. After passing directly over Jamaica on 12 September, radius, while the outer tangential wind maximum was Gilbert intensi®ed rapidly and at 2152 UTC on 13 Sep- 49±52 m sϪ1 at 61±67-km radius. Detailed descriptions tember reached a minimum sea level pressure of 888 of Hurricane Gilbert are given by Black and Willoughby mb, the lowest yet recorded in the Atlantic basin (Wil- (1992), Samsury and Zipser (1995), and Dodge et al. loughby et al. 1989). Approximately 12 h later, when (1999). its central pressure was 892 mb, the horizontal structure Echo-free moats such as the one shown in Fig. 1 are of the radar re¯ectivity and the radial pro®les of tan- often found in intense storms, even when no well-de- gential wind, angular velocity, and relative vorticity ®ned outer eyewall is present. In general, echo-free moats are regions of strong differential rotation.1 For
Corresponding author address: James P. Kossin, Department of Atmospheric Science, Colorado State University, Fort Collins, CO 80523. 1 We de®ne differential rotation as d/dr, where is the azimuthal E-mail: [email protected] mean angular velocity.
᭧ 2000 American Meteorological Society 3893 3894 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 57
FIG. 1. (a) Composite horizontal radar re¯ectivity of Hurricane Gilbert for 0959±1025 UTC 14 Sep 1988; the domain is 360 km ϫ 360 km, with tick marks every 36 km. The line through the center is the WP-3D aircraft ¯ight track (from Samsury and Zipser 1995). (b) Pro®les of ¯ight-level angular velocity (solid), tangential wind (short dash), and smoothed relative vorticity (long dash) along the southern leg of the ¯ight track shown in (a). 15 DECEMBER 2000 KOSSIN ET AL. 3895 example, in the Gilbert case, the circuit time (i.e., the axisymmetric ¯ow ®eld with four distinct regions of time required to traverse a complete circle) for a parcel vorticity: an inner region (r Ͻ r1 ഠ 10 km) of very Ϫ1 Ϫ4 Ϫ1 at a radius of 10 km, with tangential wind 68 m s ,is high vorticity 1 ഠ 159 ϫ 10 s , a moat region (r1 15.4 min, while the circuit time for a parcel at a radius Ͻ r Ͻ r 2 ഠ 55 km) of relatively low vorticity 2 ഠ Ϫ1 Ϫ4 Ϫ1 of 50 km, with tangential wind 30 m s , is 175 min. 5 ϫ 10 s , an annular ring (r 2 Ͻ r Ͻ r 3 ഠ 100 Ϫ4 Ϫ1 In other words, the parcel at 10-km radius completes km) of elevated vorticity 3 ഠ 27 ϫ 10 s , and the 11 circuits in the time the parcel at 50-km radius com- far ®eld (r Ͼ r 3 ) nearly irrotational ¯ow. The as- pletes one circuit. Under such strong differential rota- sumption of a monotonic pro®le near the vortex center tion, small asymmetric regions of enhanced potential removes the possibility for primary eyewall instabil- vorticity are rapidly ®lamented to small radial scales ities, which were the focus of Schubert et al. (1999). (Carr and Williams 1989; Sutyrin 1989; Smith and In this idealization, vorticity gradients and associated Montgomery 1995) as they become more axisymmetric. vortex Rossby waves are concentrated at the radii r1 , Thus, the moat is typically a region of active potential r 2 , r 3 . In this case, there are two types of instabilities, enstrophy cascade to small scales. In contrast, the region as the vortex Rossby wave on the positive radial vor- just inside the secondary wind maximum has weak dif- ticity gradient at r 2 can interact with either of the ferential rotation. For example, the circuit time for a vortex Rossby waves on the negative radial vorticity Ϫ1 parcel at 61-km radius, with tangential wind 52 m s , gradients at r1 and r 3 . is 123 min, compared to the previously computed 175 In the ®rst type of instability (called type 1), the min at 50-km radius. Hence, the region between 50 and dominant interactions occur between the waves asso- 61 km is characterized by weak differential rotation and ciated with r 2 and r 3 . The central vorticity does not can be considered a local haven against the ravages of play a direct role in this instability, because the vor- potential enstrophy cascade to small scales. Similarly, ticity wave at r1 is dynamically inactive. However, the the updraft cores within the primary eyewall are typi- central vorticity does induce a differential rotation be- cally embedded in the local minimum of differential tween r 2 and r 3 , and this differential rotation can help rotation that lies just inside the radius of maximum suppress the instability across the ring of elevated vor- wind. This is evident in Fig. 1b, which shows a ¯at- ticity between r 2 and r 3 (Dritschel 1989). Type 1 in- tening of the angular velocity pro®le inside the wind stability leads to a roll-up of the annular ring and the maximum. formation of coherent vorticity structures. Once roll- The tendency for convection to be suppressed in the up has occured, the ¯ow evolution is described by a moat region is often attributed to mesoscale subsidence collection of vortex merger events in which the central between two regions of strong upward motion. Dodge vortex is victorious (Melander et al. 1987b; Dritschel et al. (1999) found that the moat of Hurricane Gilbert and Waugh 1992) in the sense that the vorticity within the central vortex remains largely unchanged while the consisted of stratiform precipitation with weak (less relatively weak coherent vortices become rapidly ®- than 1 m sϪ1 ) downward motion below the bright band lamented and axisymmetrized by the differential ro- observed near 5-km height, and weak upward motion tation imposed across the moat by the intense central above. An additional mechanism for suppressed con- vortex. The result is a widening of the annular ring of vection in the moat may be strong differential rotation. elevated vorticity and a weakening but ultimate main- For example, imagine a circular 5-km-diameter cloud tenance of the secondary wind maximum. This type of updraft that lies between 15- and 20-km radius from evolution is discussed in section 2. the hurricane center. For the Gilbert wind ®eld, the In the second type of instability (called type 2), the inner edge of this updraft would be advected azimuth- dominant interactions occur between the waves asso- ally 140Њ in 10 min, while the outer edge is advected ciated with r and r , that is, across the moat. In this Ϫ1 1 2 70Њ in 10 min. Since a parcel rising at 5 m s ascends case, type 1 instabilities are largely or completely sup- only 3 km in this 10-min interval, one could imagine pressed by the presence of the central vortex. Type 2 that ordinary cumulonimbus convection embedded in instability leads to a rearrangement of the low vorticity such a ¯ow would be inhibited from persisting as the of the moat. One possible nonlinear outcome of this convection becomes increasingly susceptible to en- instability is the production of a vortex tripole in which trainment. the low vorticity of the moat pools into two satellites Concentric eyewall structures and tangential wind of an elliptically deformed central vortex, with the pro®les like those shown in Fig. 1 have been docu- whole structure rotating cyclonically. This type of evo- mented in a number of hurricanes (Samsury and Zip- lution is discussed in section 3. ser 1995; Willoughby et al. 1982, and references therein) and raise interesting questions about the dy- 2. Stabilization of an annular ring of vorticity by namic stability of hurricane ¯ows. The answers to a strong central vortex such questions require studies using a hierarchy of dynamical models, the simplest of which is the non- a. Linear stability analysis divergent barotropic model. In such a model, Hurri- The cyclonic shear zone associated with a secondary cane Gilbert, for example, might be idealized as an eyewall can be envisaged as an annular ring of uni- 3896 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 57 formly high vorticity, with large radial vorticity gra- The system (3) can be regarded as a concise mathe- dients on its edges. On the inner edge of the annular matical description of the interaction between two coun- ring the vorticity increases with radius, while on the terpropagating vortex Rossby edge waves along r 2 and outer edge the vorticity decreases with radius. In terms r3 and in¯uenced by the central vortex. As we shall see of vortex Rossby wave theory, waves on the inner edge below, with a basic ¯ow satisfying 2 Ͻ 3, the waves of the annular ring will prograde relative to the ¯ow can phase lock, and instability is possible. Note that the there, while waves on the outer edge will retrograde effects of the central point vortex enter through 2 and relative to the ¯ow there. It is possible for these two 3. The effect of a strong central vortex is to make 2 counterpropagating (relative to the tangential ¯ow in Ͼ 3, which disrupts the ability of the two waves to their vicinity) waves to have the same angular velocity phase lock. To quantify the effects of the central point relative to the earth, that is, to be phase locked. If the vortex on the stability of the annular ring, let us now locked phase is favorable, each wave will make the examine the formula for the eigenvalues . other grow, and barotropic instability will result. The The eigenvalues of (3), normalized by 3, are given by presence of a central region of high vorticity compli- cates this picture in two ways. First, the edge of the 1 23ϩ ϭ m central region can also support waves, which might 332 interact with waves along the other two edges if they 1/2 are close enough (section 3). Second, even if the an- 1 Ϫ 2 r 2m Ϯ 1 ϩ m 23Ϫ 2 . (4) nular ring of elevated vorticity between r 2 and r 3 is far 2Ά[]33r · enough away from the central region that the waves along the edge of the central region do not signi®cantly In order to more easily interpret the eigenvalue relation interact, the central region of high vorticity can induce (4), it is convenient to reduce the number of adjustable an axisymmetric differential rotation across the annular parameters to two. We ®rst de®ne ␦ ϭ r 2/r3 as a measure ring and thereby stabilize the ring. Both of these effects of the width of the annular ring of vorticity and ⌫ϭ 22 can be understood as special cases of a four-region C/[3(rr32Ϫ )] as the ratio of the central point vortex model, which is discussed in the appendix. The main circulation to the secondary eyewall circulation. Ex- result of that analysis is the eigenvalue problem (A6). pressing 2/3 and 3/3 in terms of ⌫ and ␦, it can be In order to understand the stabilizing effect of differ- shown using (4) that wavenumbers one and two are ential rotation across the annular ring, we ®rst consider exponentially stable when ⌫Ͼ0, and for m Ͼ 2, sta- → bility is guaranteed when the special case where 2 ϭ 0, and where r1 0 and → 22→ 1 ϱ in such a way that r111 ϭ 2r 1 C, where ␦ 2 ⌫Ͼ . (5) C is a speci®ed constant circulation associated with the 2 central point vortex. In this special case the axisym- 1 Ϫ ␦ metric basic state angular velocity (r) given by (A2) The stability condition (5) is equivalent to Dritschel's 2 reduces to (1989) condition, ⌳Ͼ1, where ⌳ϭC/(3r2)isa dimensionless ``adverse shear.'' The region of the ⌫±␦ 00Յ r Յ r , C 1 2 plane satisfying (5) lies above the dashed lines in Figs. (r) ϭϩ 1 Ϫ (r /r)2 r Յ r Յ r , 2 and 3. A physical interpretation of the stability con- 2r 2 2 32 2 3 (r /r)22Ϫ (r /r) r Յ r Ͻϱ, dition (5) is related to Fjùrtoft's theorem (Montgomery 32 3 and Shapiro 1995) and can be understood by noting that (1) since the right-hand side of (5) is less than unity and and the corresponding basic state relative vorticity (r) the left-hand side is greater than unity if ( 2 Ϫ 3)/3 given by (A3) reduces to Ͼ 0, stability is assured if 2 Ͼ 3 (for 3 Ͼ 0). Since, in the absence of coupling, the waves on the inner edge 00Ͻ r Ͻ r , of the annular ring prograde relative to the ¯ow in its d(r 2 ) 2 (r) ϭϭ r Ͻ r Ͻ r , (2) vicinity, and since the waves on the outer edge retro- rdr 32 3 grade relative to the ¯ow in its vicinity, a larger basic- 0 r3 Ͻ r Ͻϱ, state angular velocity on the inner edge ( 2 Ͼ 3) will prevent the waves from phase locking, satisfying a suf- where r 2, r3 are speci®ed radii and 3 a speci®ed vor- ticity level. This idealized basic state was also studied ®cient condition for stability. by Dritschel (1989). The eigenvalue problem (A6) re- Wavenumbers larger than two can produce frequen- duces to cies with nonzero imaginary parts. Expressing 2 / 3 1 Ϫ2 and 3 / 3 in terms of ␦ and ⌫ as 2 / 3 ϭ⌫2 (␦ Ϫ 11 1) and / ϭ 1 (⌫ϩ1)(1 Ϫ ␦ 2 ), and then using these m ϩ (r /r )m 3 3 2 23 323 in (4), we can calculate the dimensionless complex 22 ⌿⌿22 ϭ . (3) frequency / as a function of the disturbance azi- 11⌿⌿ 3 m 33 muthal wavenumber m and the two basic-state ¯ow Ϫ 323(r /r ) m 3Ϫ 3 22 parameters ␦ and ⌫. The imaginary part of / 3 , de- 15 DECEMBER 2000 KOSSIN ET AL. 3897
22 FIG. 2. Isolines of the dimensionless growth rate i/3, computed from (4), as a function of ␦ ϭ r2/r3 and ⌫ϭC/[3(rr32Ϫ )] for azimuthal wavenumbers m ϭ 3, 9, and 16. The parameter ⌫ is the ratio of the circulation associated with the central point vortex to the circulation associated with the annular ring of elevated vorticity between r2 and r3. Nonzero growth rates occur only in the shaded regions.
The isolines are i/3 ϭ 0.01, 0.03, 0.05, ....Themaximum growth rates increase and are found closer to ␦ ϭ 1asm increases. The region above the dashed line satis®es the suf®cient condition for stability given by (5).
noted by i / 3 , is a dimensionless measure of the growth rate. Isolines of i / 3 as a function of ␦ and ⌫ for m ϭ 3, 9, and 16 are shown in Fig. 2. Note that all basic states, no matter what the value of ⌫, satisfy the Rayleigh necessary condition for instability, but that most of the region shown in Fig. 2 is in fact stable.
Clearly, thinner annular regions (larger values of r 2 /r 3 ) should produce the highest growth rates but at much higher azimuthal wavenumbers. Note also the overlap in the unstable regions of the ⌫±␦ plane for different azimuthal wavenumbers. For example, the lower right area of the ⌫±␦ plane is unstable to all the azimuthal wavenumbers m ϭ 3, 9, and 16. We can combine the three panels in Fig. 2 with the remaining growth rate plots for m ϭ 3,4,...,16andcollapse them into a single diagram if, for each point in the ⌫±␦ plane, we choose the largest growth rate of the fourteen wav- enumbers. This results in Fig. 3. To estimate the growth rates expected in a secondary eyewall mixing problem, consider the cases ␦ ഠ 0.84 and ⌫ ഠ 0.45, which are suggested by the Hurricane Gilbert data shown in Fig. 1 (but for the special case
of 2 ϭ 0). Then, from the isolines drawn in Fig. 3, we Ϫ3 obtain i ഠ 0.123 for m ϭ 8. Using 3 ഠ 2.8 ϫ 10 Ϫ1 Ϫ1 s , we obtain an e-folding time i ഠ 50 min. The
analysis of section 2c, where the restriction that 2 ϭ 0 is relaxed, results in a maximum instability for m ϭ 9.
FIG. 3. Isolines of the maximum dimensionless growth rate i/3 among the azimuthal wavenumbers m ϭ 3,4,..., 16for type 1 b. Nondivergent barotropic spectral model instability. The isolines are the same as in Fig. 2. Shading indicates the wavenumber associated with the maximum dimensionless growth To isolate the barotropic aspects of the nonlinear evo- rate at each point. lution, we now consider the nondivergent barotropic 3898 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 57 model with spectral discretization and ordinary diffu- environment to form a broader and weaker vorticity sion. Expressing the velocity components2 in terms of ring. All of the solutions presented in this paper were x, we obtained with a double Fourier pseudospectral codeץ/ץy and ϭץ/ץthe streamfunction by u ϭϪ can write the vorticity equation as having 1024 ϫ 1024 equally spaced collocation points on a doubly periodic domain of size 600 km ϫ 600 (, )ץ ץ ϩϭٌ2, (6) km. The code was run with a dealiased calculation of x, y) the quadratic nonlinear terms in (6), resulting in 340)ץ tץ where ϫ 340 resolved Fourier modes. Although the collo- cation points are only 0.585 km apart, a more realistic 2 ϭ (7) estimate of resolution is the wavelength of the highest ٌ Fourier mode, which is 1.76 km. Time differencing is the invertibility principle. Two integral properties as- was accomplished with a standard fourth-order Run- sociated with (6) and (7) on a closed or periodic domain ge±Kutta scheme. For the experiment of section 2c, are the energy and enstrophy relations the time step used was 7.5 s and the chosen value of dE viscosity in (6) was ϭ 4m2 sϪ1 , resulting in a 1/e ϭϪ2Z, (8) dt damping time of 5.5 h for all modes having total wavenumber 340. However, for modes having total dZ wavenumber 170, the damping time lengthens con- ϭϪ2P, (9) dt siderably to 22 h. For the experiment of section 2d, the time step used was 15 s and the chosen value of 1 2 Ϫ1 ١ dx dy is the energy, Z ϭ viscosity was ϭ 32 m s . This gives 1/e damping ´ ١ where E ϭ ## 2 112 ١ dx times of 41 min and 3 h for all modes having total ´ ١ ## 22 dx dy is the enstrophy, and P ϭ ## dy is the palinstrophy. The diffusion term on the right- wavenumber 340 and 170, respectively. The larger hand side of (6) controls the spectral blocking asso- value of viscosity used in the second experiment was ciated with the enstrophy cascade to higher waven- chosen in response to the more vigorous mixing and umbers. associated palinstrophy production and enstrophy cas- We now present numerical integrations of (6)±(7) cade that occurs during the integration. that demonstrate the process by which an unstable As the initial condition for (6), we use (r, , 0) ϭ outer ring of enhanced vorticity mixes with its near (r) ϩ Ј(r, ) where
1,0Յ r Յ r11Ϫ d
1S[(r Ϫ r 111ϩ d )/2d ] ϩ 211S[(r ϩ d Ϫ r)/2d 1], r 11Ϫ d Յ r Յ r 11ϩ d
2, r11ϩ d Յ r Յ r 22Ϫ d
2S[(r Ϫ r 222ϩ d )/2d ] ϩ 322S[(r ϩ d Ϫ r)/2d 2], r 22Ϫ d Յ r Յ r 22ϩ d (r) ϭ 3, r22ϩ d Յ r Յ r 33Ϫ d (11)
3S[(r Ϫ r 333ϩ d )/2d ] ϩ 433S[(r ϩ d Ϫ r)/2d 3], r 33Ϫ d Յ r Յ r 33ϩ d
4, r33ϩ d Յ r Յ r 44Ϫ d
4S[(r Ϫ r 444ϩ d )/2d ] ϩ 544S[(r ϩ d Ϫ r)/2d 4], r 44Ϫ d Յ r Յ r 44ϩ d
5, r44ϩ d Յ r
2 3 is a circular four-region vorticity distribution and r1, r 2, Here S(s) ϭ 1 Ϫ 3s ϩ 2s is the basic cubic Hermite r3, r 4, d1, d 2, d3, d 4, 1, 2, 3, 4 are independently shape function satisfying S(0) ϭ 1, S(1) ϭ 0, SЈ(0) ϭ speci®ed quantities. The constant 5 will be determined SЈ(1) ϭ 0. Since 5 must generally be weakly negative in order to make the domain average of (r, , 0) vanish. to satisfy the zero circulation requirement, but no neg- ative vorticity is expected in the region of hurricanes,
we apply a small but positive ``far ®eld'' vorticity 4 to a radius beyond the area of expected mixing processes. 2 Throughout this paper the symbols u, are used to denote east- For the experiments of sections 2c and 2d, we apply an ward and northward components of velocity when working in Car- tesian coordinates and to denote radial and tangential components azimuthally broadband perturbation across the annular when working in cylindrical coordinates. region r2 Ϫ d 2 Յ r Յ r3 ϩ d3, that is, 15 DECEMBER 2000 KOSSIN ET AL. 3899
0, 0 Յ r Յ r22Ϫ d S[(r d r)/2d ], r d Յ r Յ r d 22ϩ Ϫ 2 22Ϫ 22ϩ 12 Ј(r, ) ϭ cos(m)1, r ϩ d Յ r Յ r Ϫ d (12) amp 2 2 3 3 mϭ1 S[(r Ϫ r ϩ d )/2d ], r Ϫ d Յ r Յ r ϩ d 333 33 33
0, r33ϩ d Յ r Ͻϱ
where amp is a speci®ed constant less than 1% of the sociated e-folding time of 67 minutes (solid line of maximum vorticity in the annulus. Fig. 5). Results of this experiment are shown in Fig. 6 in the form of vorticity maps plotted over a 24-h period. c. Type 1 instability in the presence of a central The growth of the wavenumber-nine maximum insta- vortex: Maintenance of a secondary wind bility becomes evident within 4 h and by t ϭ 6 h the maximum ring has undergone a nearly complete roll-up into nine coherent structures. The differential rotation induced An initial condition that simulates an annular ring by the central vortex advects the inner edges of the of high potential vorticity (PV) with concentrated vor- coherent structures more rapidly than the outer edges ticity at its center may be imposed by choosing k 1 so that at t ϭ 8 h, the structures have become cyclon- K 3 Ͼ 2 and r1 r 2 in (11). In this way, wave inter- ically stretched around the vortex, while trailing spirals actions between the ring and the central vortex are have formed as vorticity is stripped from their outer minimized and the ring ``feels'' the presence of the edges. At t ϭ 10 h, the stretching has resulted in a central vortex only through the angular velocity ®eld. banded structure with thin strips of enhanced vorticity For the ®rst experiment, the numerical integration is being wrapped around the vortex. Leading spirals, performed under the initial condition {r1 , r 2 , r 3 , r 4 } which have been stripped from the inner edges of the ϭ {9.5, 52.5, 62.5, 120.0} km, {d1 , d 2 , d 3 , d 4 } ϭ {2.5, bands, have propagated inward into regions of intense 2.5, 2.5, 15.0} km, and {1 , 2 , 3 , 4 , 5 } ϭ {159.18, differential rotation (and active enstrophy cascade). At 5.18, 27.18, 2.18, Ϫ0.82} ϫ 10Ϫ4 sϪ1 . The associated t ϭ 12 h, the moat has been maintained, although its tangential wind pro®le is analogous to an observed outer radius has contracted to 35 km from its initial pro®le from a single National Oceanic and Atmospher- value of 60 km. Outside r ϭ 35 km, the vorticity of ic Administration (NOAA) WP-3D radial ¯ight leg into the strips is suf®ciently strong to maintain the strips Hurricane Gilbert during 1012±1029 UTC 14 Septem- against the vortex-induced differential rotation. Inside ber 1988 (as shown in Fig. 1). Radial pro®les of the r ϭ 35 km, the strips can no longer maintain them- symmetric part of the initial vorticity, tangential wind, selves and they are rapidly ®lamented to small scales and angular velocity ®elds are shown by the solid lines where they are lost to diffusion. Thus for these initial in Fig. 4 (the other lines in this ®gure will be discussed conditions, r ϭ 35 km represents a barrier to inward later). The maximum wind of 69 m sϪ1 is found near mixing. Also evident at this time, particularly in the 10 km, while a secondary maximum of 47 m sϪ1 , as- two eastern quadrants of the vortex, are secondary in- sociated with the ring of enhanced vorticity, is evident stabilities across individual strips of vorticity. These near 64 km. The prograding vortex Rossby waves instabilities occur as the strips become thinner and are along the inner edge of the ring (50 Յ r Յ 55 km) are most pronounced at larger radii where the vortex-in- embedded in a local angular velocity of approximately duced differential rotation is weak. The occurence of 5.6 ϫ 10Ϫ4 sϪ1 , while the retrograding waves along the secondary instabilities in our experiment can be related outer edge of the ring (60 Յ r Յ 65 km) are embedded to the results of Dritschel (1989) by estimating the 2 in a stronger local angular velocity of approximately adverse shear ⌳ϭC/(3r2) across these strips. Using Ϫ4 Ϫ1 2 7.3 ϫ 10 s . Although the presence of the central C ഠ 1r1 , we obtain ⌳ ഠ 0.187. For the case of a vortex reduces the differential rotation across the an- linear shear ¯ow, Dritschel found that such secondary nular ring by about 30% when compared to the dif- instabilities lead to a roll-up into a string of vortices ferential rotation with the central vortex removed, it is when ⌳Ͻ0.21, while for larger ⌳, the adverse shear not enough to eliminate all instabilities. A piecewise inhibits this secondary roll-up even though the strips uniform vorticity pro®le as described in the four-region are in fact unstable. We see then that the presence of model of the appendix and that imitates the smooth a central vortex in¯uences the ¯ow evolution in two initial vorticity pro®le of Fig. 4 is found to be unstable somewhat disparate ways. In the region of the moat, in wavenumbers seven through 10, with the maximum coherent vorticity structures are ®lamented and lost to growth rate occuring in wavenumber nine with an as- diffusion, while at larger radii (but still inside r ϭ 80 3900 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 57
FIG. 4. (a) Azimuthal mean vorticity and tangential velocity, and (b) angular velocity for the experiment shown in Fig. 6 at the selected times t ϭ 0 (solid), 6 h (long dash), 12 h (medium dash), and 24 h (short dash). Averages were computed with respect to distance from the minimum streamfunction position. The maximum angular velocity is truncated in the image to highlight the region of the annular ring. Note the reversal of differential rotation across the ring between 6 and 12 h. km), the vortex helps to maintain the vorticity by in- pro®les represent azimuthal averages taken with respect hibiting the growth of secondary instabilities across to distance from the minimum streamfunction.3 At t ϭ the thin strips. 6 h, the annular ring has become broader and weaker At t ϭ 15 h, the instabilities across the strips have and its maximum vorticity has moved inward. The tan- rolled up into miniature vortices and during the re- gential wind pro®le shows a weakening of the secondary mainder of the evolution to t ϭ 24 h, these vortices maximum while the ¯ow inside of the ring has strength- undergo mixing and merger processes, relaxing the ¯ow ened, smoothing out the initially steep gradient between toward axisymmetry. Further time integration past t ϭ 24 h results in little appreciable change other than a slow diffusive spindown. 3 Although the centroid of vorticity on an f plane is invariant, The dashed lines of Fig. 4 show the evolution of the asymmetries formed during the evolution cause the location of the vorticity, tangential velocity, and angular velocity. The minimum streamfunction to oscillate. 15 DECEMBER 2000 KOSSIN ET AL. 3901
FIG. 5. Dimensional growth rates and e-folding times computed from Eq. (A6) for the piecewise uniform approximations to the initial conditions of sections 2c and 2d. The dots correspond to the integer values of azimuthal wavenumber m.
52 and 62 km. The angular velocity evolution exhibits stable secondary eyewall is just enough to maintain a a lessening of the differential rotation across the ring. near-neutral ¯ow. At t ϭ 12 h, the symmetric part of the ¯ow has relaxed to a nearly steady state. The vorticity in the ring has further broadened and weakened but the secondary max- d. Instability in the absence of a central vortex: imum and its associated gradient reversal have persisted. Redistribution into a monopole Similarly, the tangential wind maintains a signi®cant For comparison, a numerical simulation is performed secondary maximum with the ¯ow increasing from 30 with the central vortex removed, so that 1 ϭ 2 ϭ 5.18 msϪ1 at 35 km to 42 m sϪ1 at 73 km. The angular ϫ 10Ϫ4 sϪ1 and all other parameters in (11) remain es- velocity has become monotonic, eliminating the positive sentially unchanged.4 For further discussion of the evo- differential rotation across the ring. Thus the symmetric lution of unstable rings of enhanced vorticity without part of the ¯ow has become stable in the context of the the presence of a central vortex, the reader is directed four-region model, and little change is noted during the to Schubert et al. (1999). Radial pro®les of the initial next 12 h. vorticity, tangential wind, and angular velocity ®elds The stability of the nearly equilibrated ¯ow of the are shown by the solid lines in Fig. 7. The differential above experiment may offer insight into the observed rotation across the ring is greater than the previous ex- longevity of hurricane secondary eyewalls. Our results ample ( ഠ 3.1 ϫ 10Ϫ4 sϪ1 when 50 Յ r Յ 55 km and are based on an initial condition that imitates a pro®le ഠ 5.3 ϫ 10Ϫ4 sϪ1 when 60 Յ r Յ 65 km) and the observed during a single radial ¯ight leg into Hurricane maximum growth rate has shifted from wavenumber Gilbert on 14 September 1988. This pro®le was chosen nine to six while the e-folding time has decreased from to demonstrate the mechanism whereby PV mixing can 67 to 48 min (dashed line in Fig. 5). The evolution is occur while still maintaining a secondary wind maxi- again displayed in the form of vorticity maps in Fig. 8. mum. Black and Willoughby (1992) calculated azi- At t ϭ 4 h, the wavenumber-six maximum instability muthal mean tangential wind pro®les in Hurricane Gil- has ampli®ed and a roll-up of the annular ring has oc- bert based on aircraft observations at the beginning and cured. The presence of the wavenumber-seven instabil- end of the 14 September sortie (shown in their Fig. 4c). ity, whose growth rate is 96% of the maximum insta- In the context of the four-region model, the mean pro®le bility, is evident in the northern quadrant. At t ϭ 8h, at the beginning was found to be stable but near neutral, six well formed elliptical vortices have emerged. Weak and the mean pro®le based on the ¯ow near the end of vorticity from outside the ring is beginning to enter the the sortie was stable. Thus, the results of the experiment shown in Fig. 6 are useful in describing how an unstable secondary eyewall can evolve to a stable secondary eye- 4 The zero-circulation requirement of the doubly periodic domain wall, but these results could likely be an exaggeration results in a shift in the vorticity everywhere when any local change of the actual evolution of a hurricane's secondary eye- is made. Removing the central vortex causes a slight shift in 2, 3, wall. It is possible that mixing associated with an un- 4, 5. 3902 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 57
FIG. 6. Vorticity contour plots for the type 1 instability experiment. The model domain is 600 km ϫ 600 km but only the inner 200 km ϫ 200 km is shown. Successively darker shading denotes successively higher vorticity. Initially, high central vorticity is shaded black, enhanced vorticity in the ring is dark gray, and low vorticity in the moat and outside the eyewall is white. (a) t ϭ 0tot ϭ 8h.(b)t ϭ 10htot ϭ 24 h. The time interval switches from 2 to 3 h after t ϭ 12 h. 15 DECEMBER 2000 KOSSIN ET AL. 3903
FIG.6.(Continued) 3904 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 57
nopolizes the in¯ow of moist energy, the primary eye- wall typically spins down. Both processes act to desta- bilize the secondary eyewall and result in asymmetric mixing that moves the inner edge of the secondary eye- wall inward while the ¯ow tends toward stability. As the secondary eyewall restrengthens, instabilities again emerge and the contraction process continues. As the primary eyewall weakens, the barrier to inward mixing is reduced until the mixing can reach the center, thus completing the cycle. In this case, the mechanisms that strengthen the secondary eyewall and weaken the pri- mary eyewall rely largely on the divergent circulation but the contraction and ultimate replacement of the pri- mary eyewall can be explained in terms of nondivergent asymmetric mixing processes alone.
3. Formation and persistence of an elliptical central vortex Tropical cyclones sometimes have rotating elliptical eyes. Two outstanding examples were recently docu- mented by Kuo et al. (1999) for the case of Typhoon Herb (1996), and by Reasor et al. (2000) for the case of Hurricane Olivia (1994). Typhoon Herb was observed using the WSR-88D radar on Wu-Feng Mountain in Taiwan. The elliptical eye of Typhoon Herb had an as- pect ratio (major axis/minor axis) of approximately 1.5. Two complete rotations of the elliptical eye, each with FIG. 7. Similar to Fig. 4 but for the experiment with the central a period of 144 min, were observed as the typhoon vortex removed (shown in Fig. 8). The azimuthal mean vorticity approached the radar. Kuo et al. interpreted this elliptical pro®le when t ϭ 54 h is monotonic. eye in terms of the Kirchhoff vortex (Lamb 1932, p. 232), an exact, stable (for aspect ratios less than 3) solution for two-dimensional incompressible ¯ow hav- central region of the vortex, while stronger vorticity ing an elliptical patch of constant vorticity ¯uid sur- from the center is ejected outward in the form of trailing rounded by irrotational ¯ow. Is the Kirchhoff vortex, spirals. After this time, intricate vortex merger and mix- with its discontinuous vorticity at the edge of the ellipse, ing processes dominate the evolution so that at t ϭ 16 a useful model of rotating elliptical eyes? Melander et h, ®ve vortices remain, and at t ϭ 24 h, only three al. (1987a) have shown that elliptical vortices with remain. Without the presence of a dominant central vor- smooth transitions to the far-®eld irrotational ¯ow are tex, each coherent structure imposes a similar ¯ow not robust but tend to become axisymmetrized via an across the others. There is no clear victor in the merger inviscid process in which a halo of vorticity ®laments process and high vorticity can mix to the center. At t become wrapped around a more symmetric vortex core. ϭ 54 h, the vorticity has nearly relaxed to a monotonic Montgomery and Kallenbach (1997) further clari®ed the distribution that satis®es the Rayleigh suf®cient con- physics of this process for initially monopolar distri- dition for stability. This behavior is in marked contrast butions of basic-state vorticity, showing that axisym- to the behavior in the presence of a central vortex (sec- metrization can be described as the radial and azimuthal tion 2c), where the mixing results in a monotonic an- dispersion of vortex Rossby waves that are progres- gular velocity distribution which satis®es Fjùrtoft's, but sively sheared by the differential rotation of the vortex not Rayleigh's, suf®cient condition for stability across winds. In this latter work, the dependence of the local the ring. wave frequency and stagnation radius on the radial vor- The vorticity rearrangement associated with type 1 ticity gradient was made explicit. In other recent work, instability suggests a nondivergent mechanism for the Koumoutsakos (1997) and Dritschel (1998) have shown contraction of secondary eyewalls that may augment the that the extent to which a vortex axisymmetrizes is con- axisymmetric contraction mechanism discussed by Sha- trolled by the steepness of the vortex edge, with very piro and Willoughby (1982) and Willoughby et al. sharp initial vorticity distributions resulting in robust (1982). As part of an eyewall replacement cycle, the nonaxisymmetric con®gurations. This result is congru- secondary eyewall strengthens through convergent and ent with Kelvin's linear analysis for Rossby edge waves, convective processes and as the secondary eyewall mo- in which the asymmetries never axisymmetrize on a 15 DECEMBER 2000 KOSSIN ET AL. 3905
FIG. 8. Similar to the experiment of Fig. 6 but without the presence of the central vortex. The shading is the same as in Fig. 6. Plots are for the times t ϭ 0, 4, 8, 16, 24, and 54 h. 3906 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 57 discontinuous basic-state vorticity pro®le. If we now tude asymmetric perturbations applied to a stable mono- allow a sharp but smooth gradient, the local Wenzel± pole do not always relax to their axisymmetric base Kramers±Brillouin (WKB) theory of Montgomery and state. The generation mechanism of probable impor- Kallenbach (1997) suggests radially trapped waves that tance to tropical cyclone dynamics, however, is that as- axisymmetrize very slowly. The physics is generally a sociated with the barotropic instability of axisymmetric struggle between Rossby elasticity and radial shearing. shielded or, more importantly, partially shielded5 vor- Dritschel (1998, Figs. 6 and 7) gives an example of a tices (Gent and McWilliams 1986; Flierl 1988). Here robust, quasi-steady rotating elliptical pattern of vortic- we consider the central core of the vortex to be mono- ity with aspect ratio 1.6 in which approximately 40% tonic and, hence, eliminate possibly dominant instabil- of the vorticity drop from the center to the far ®eld ities that may occur across the eyewall, as investigated occurs discontinuously at the edge of the ellipse. A re- by Reasor et al. (2000). cent study by Montgomery and Enagonio (1998) further suggests that even in the absence of any discontinuity a. Instability across the moat of a shielded vortex in a monotonic basic-state vorticity pro®le, convectively generated vortex Rossby waves can modify the basic A well-studied family of shielded monopoles has the state and cause a reversal in radial vorticity pro®le. The angular velocity (r) given by sign change allows for the existence of a discrete neutral (r) ϭ exp[Ϫ(r/b)␣], (13) or weakly unstable vortex Rossby mode that may not 0 decay. where 0 is the angular velocity at r ϭ 0, b the size of Reasor et al. (2000) documented the elliptical eyewall the vortex, and ␣ the steepness parameter, with ␣ Ͼ 0. of Hurricane Olivia using NOAA WP-3D airborne dual- The associated tangential wind (r) ϭ r(r) and vor- Doppler radar data and ¯ight-level data, and identi®ed ticity (r) ϭ d(r)/rdr are given by barotropic instability across the eyewall as a possible (r) ϭ r exp[Ϫ(r/b)␣ ], (14) source for the observed wavenumber-2 asymmetry. 0 During the observation period, the symmetric part of 1 ␣␣ the vorticity consisted of an elevated annular region (r) ϭ 20 1 Ϫ ␣(r/b) exp[Ϫ(r/b) ]. (15) within the eyewall surrounding a depressed region with- []2 in the eye. An associated smooth basic-state vorticity Plots of / 0, /(b0), and /(20) as functions of r/b pro®le was found to support instabilities with a maxi- for ␣ ϭ 2, 3, 4, 5, 6 are shown in Fig. 9. The radius mum growth in wavenumber two. The observed wave- of maximum tangential wind occurs at r/b ϭ (1/␣)1/␣, number-2 vorticity asymmetry in the eyewall was hy- and the vorticity reverses sign at r/b ϭ (2/␣)1/␣ and is pothesized to be associated with the breakdown of an a minimum at r/b ϭ (1 ϩ 2/␣)1/␣. Note that for increas- initially unstable ring of elevated vorticity as described ing ␣, the vorticity in the core becomes more uniform, by Schubert et al. (1999) and in section 2d of this study. the annulus of negative /(2 0) becomes thinner and An alternative interpretation of the elliptical patterns the vorticity gradient at the edge of the core becomes observed in hurricanes is that their vorticity ®elds had steeper. The instability and nonlinear evolution of this evolved into structures resembling tripoles. Strictly particular initial vorticity pro®le has been studied by speaking, a tripole is de®ned as a linear arrangement of Carton and McWilliams (1989), Carton et al. (1989), three regions of distributed vorticity of alternate signs, Orlandi and van Heijst (1992), Carton and Legras with the whole con®guration steadily rotating in the (1994), and Carnevale and Kloosterziel (1994). Carton same sense as the vorticity of the elliptically shaped and McWilliams (1989) showed that this vortex is lin- central core. Tripoles have been the subject of several early stable for ␣ Շ 1.9 and unstable for larger values laboratory experiments (Kloosterziel and van Heijst of ␣. Carnevale and Kloosterziel (1994, their Fig. 7) 1991; van Heijst et al. 1991; Denoix et al. 1994) and showed that wavenumber two remains the fastest grow- numerical simulations ever since they were shown to ing wave until ␣ ഠ 6, at which point wavenumber three emerge as coherent structures in forced two-dimensional becomes the fastest growing wave. Within the range 1.9 turbulence simulations (Legras et al. 1988). In addition Ͻ ␣ Ͻ 6, the wavenumber-two instability can saturate to their emergence as coherent structures in two-di- as two distinct quasi-stable structures. Carton and Le- mensional turbulence, tripoles can be produced in a va- gras (1994) found that when ␣ Ͻ 3.2, saturation to a riety of ways, some quite exotic from the point of view tripole occurs, while for ␣ Ͼ 3.2, two separating dipoles of tropical cyclone dynamics. For example, tripoles can emerge. As ␣ is increased, the nonlinear ampli®cation be produced by the collision of two asymmetric dipoles (Larichev and Reznik 1983) or by the offset collision of two symmetric Lamb dipoles (Orlandi and van Heijst 1992). Rossi et al. (1997) applied a quadrupolar dis- 5 An annular ring of negative vorticity can ``partially shield'' the far-®eld ¯ow from a central core of positive vorticity just as, for tortion to a monopolar Gaussian vorticity distribution example, in the element lithium (atomic number 3), the two electrons and found that for a large enough distortion, relaxation that occupy the inner, spherical 1s-orbital ``partially shield'' the out- to a tripole can occur, demonstrating that ®nite-ampli- ermost electron from the ϩ3 charge of the nucleus. 15 DECEMBER 2000 KOSSIN ET AL. 3907
FIG. 9. Pro®les of dimensionless angular velocity, tangential wind, and vorticity for unstable shielded monopoles. The maximum growth FIG. 10. (a) Saturation of a wavenumber-2 maximum instability to is in m ϭ 2 when ␣ ϭ 2, 3, 4, 5 and m ϭ 3 when ␣ ϭ 6. The pro®le a tripole for a shielded monopole with ␣ ϭ 3. The contours begin at Ϫ4 Ϫ1 Ϫ4 Ϫ1 is stable when ␣ Յ 1.9. Ϫ7 ϫ 10 s and are incremented by 5 ϫ 10 s . (b) Saturation to two separating dipoles when ␣ ϭ 4. The contours begin at Ϫ15 ϫ 10Ϫ4 sϪ1 and are incremented by 6 ϫ 10Ϫ4 sϪ1. The model domain for both plots is 600 km ϫ 600 km, and values along the label bar are in units of 10Ϫ4 sϪ1. of wavenumber two increasingly elongates the central monopole, and when ␣ Ͼ 3.2, the central monopole is elongated to the point where it breaks as the two dipoles separate. as the negative vorticity of the annulus is anticycloni- Two experiments are now performed using initial vor- cally wrapped into two satellite vortices. The anticy- ticity pro®les given by (15) with b ϭ 35 km and 0 ϭ clonic advection of the vorticity originally in the annulus 1.85 ϫ 10Ϫ3 sϪ1. For the ®rst experiment, ␣ ϭ 3 and is not due to the sign of the vorticity there, but is due for the second, ␣ ϭ 4. In both experiments, an initial to the arrangement of the positive vorticity being pulled perturbation of proportional (1%) random noise is im- from the central region. This will be more clearly dem- posed across the moat. onstrated in section 3c. During the later stages, the as- When ␣ ϭ 3, the wavenumber-two maximum insta- pect ratio (major axis/minor axis) of the elliptical central bility results in a rearrangement of the vorticity into a region has decreased to 1.7 with the satellite vortices tripole (Fig. 10a). During the early stages of the evo- found along the minor axis. lution, the central vorticity becomes highly elongated When ␣ ϭ 4, the end result of the vorticity rear- 3908 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 57 rangement is dramatically different. The wavenumber- Analogously to the discussion of section 2a, the system two maximum instability does not saturate as a tripole (18) describes the interaction between two counterpro- but results in the formation of two separating dipoles pagating vortex Rossby waves with the wave along the
(Fig. 10b). In the early part of the evolution, the negative edge of the vortex (r ϭ r1) propagating clockwise rel- vorticity within the moat is strong enough to elongate ative to strong cyclonic ¯ow, and the wave along the the central region into a strip and two pools of positive outer edge of the moat (r ϭ r 2) propagating counter- vorticity form within the strip near the two satellite vor- clockwise relative to weaker ¯ow. In this case, both the tices. The strip is eventually broken as the two dipoles Rayleigh and Fjùrtoft necessary conditions for insta- propagate away from each other. bility are satis®ed. The eigenvalues of (18), normalized
Since the radius of minimum vorticity is given by r/b by 1, are given by ϭ (1 ϩ 2/␣)1/␣, the evaluation of (15) at this radius 1 2 yields the minimum vorticity min ϭϪ 0␣ exp[Ϫ(1 ϩ ϭ {m[1 ϩ / ϩ (1 Ϫ / )(r /r )]Ϫ 1} 4 21 21 12 2/␣)], from which it is easily shown that f ϩ min Ͻ 0 1 when 2 / f Ͼ (2/␣) exp(1 ϩ 2/␣). Thus, the minimum 0 1 22 absolute vorticity is negative when 2 0/ f Ͼ 7.39, 3.53, Ϯ͗{m(1 Ϫ 21/ )[1 Ϫ (r 12/r )]Ϫ [1 Ϫ 2( 21/ )]} 2.24 for ␣ ϭ 2, 3, 4 respectively. Recognizing that 4 tropical cyclones have values of central vorticity ex- 2m 1/2 ϩ 4(21/ )(1 Ϫ 21/ )(r 12/r ) ͘ . (19) ceeding 50 times the Coriolis parameter (i.e., 20/ f Ͼ 50), the use of (13)±(15) as an initial condition in a full Figure 11 shows isolines of the imaginary part of /1 primitive equation model would result in regions of un- as a function of r1/r2 and Ϫ 2/1. Although all basic realistic negative inertial stability, that is, regions with states satisfy both the Rayleigh and Fjùrtoft necessary ( f ϩ 2 /r)( f ϩ ) Ͻ 0. These considerations make conditions for instability, much of the region shown in application of the well-studied family of shielded mono- Fig. 11 is stable, and for any value of Ϫ2/1, instability poles (13)±(15) to tropical cyclones highly questionable. can occur only when r1/r2 տ 0.38. An interesting feature In section 3c, we will demonstrate that more realistic observed in Fig. 11 is that a larger central vortex is initial conditions, with positive vorticity in the moat, more susceptible to instability across its moat. For ex- can also lead to the formation of tripoles. ample, the secondary eyewall in Hurricane Gilbert on 14 September would need to contract to a radius less than 17 km in order for type 2 instability to occur. For b. Linear stability analysis of a partially shielded the case of a larger central vortex (r ϭ 30 km) sur- vortex 1 rounded by a secondary eyewall where Ϫ 2/1 is the Returning to the four-region model of the appendix, same as the Gilbert case, type 2 instability occurs when consider the special case 1 Ͼ 0, 2 Ͻ 0, and 3 ϭ 0. the inner edge of the secondary eyewall contracts to a The axisymmetric basic-state angular velocity (r) giv- radius of 55 km. en by (A2) then reduces to 0 Յ r Յ r , 1 11c. Type 2 instability across a moat of positive (r) ϭ [ r 222ϩ (r Ϫ r )]rrϪ2 Յ r Յ r , vorticity: Application to hurricanes 2 11 2 1 1 2 222Ϫ2 [11r ϩ 2(r 2Ϫ r 1)]rr 2Յ r Ͻϱ, The wind structure of a hurricane is intimately tied (16) to the convective ®eld, with the convection tending to produce low-level convergence and hence cyclonic vor- and the corresponding basic-state relative vorticity (r) ticity in the area of a convective ring. When convection given by (A3) reduces to is concentrated near the inner edge of a ring of enhanced 0 Ͻ r Ͻ r , vorticity, a contraction of the ring may follow in re- d(r 2 ) 11 (r) ϭϭ r Ͻ r Ͻ r , (17) sponse to nonconservative forcing (Shapiro and Wil- rdr 21 2 loughby 1982; Willoughby et al. 1982; Willoughby 0 r2 Ͻ r Ͻϱ, 1990). As the ring contracts, the differential rotation where r1, r 2 are speci®ed radii and 1, 2 speci®ed vor- imposed across the ring by the central vortex becomes ticity levels. The eigenvalue problem (A6) reduces to greater and eventually reverses the self-induced differ- ential rotation of the ring. At this point, the ring is 11 m assured to have no type 1 instabilities. Further contrac- m1212112ϩ ( Ϫ )( Ϫ )(r /r ) 22 ⌿1 tion, however, brings the inner edge of the ring closer 11⌿ to the central vortex where type 2 instabilities between m 2 Ϫ 212(r /r ) m 2Ϫ 2 the ring and central vortex can take place. Recalling 22 Fig. 11 and noting that during a contraction, r1/r2 in- creases from small values toward unity, we expect type ⌿ ϭ 1 . (18) 2 instabilities to appear ®rst with a maximum growth ⌿2 rate in wavenumber two. The vorticity mixing associ- 15 DECEMBER 2000 KOSSIN ET AL. 3909
FIG. 11. Isolines of the maximum dimensionless growth rate i/1 among the azimuthal wavenumbers m ϭ 3,4,...,16 for instability of a partially shielded vortex. The isolines and shading are the same as in Fig. 3. ated with such an instability perturbs the vortex into a vortex of uniform vorticity surrounded by a thin ring tripole and offers an explanation for the origin and per- of enhanced vorticity located at r ϭ 42.5 km or 1.7 sistence of elliptical eyewalls in hurricanes. times the radius of maximum wind. The symmetric part A numerical integration is now performed under the of the initial vorticity and tangential wind ®elds are initial condition parameters {r1, r 2, r3, r 4} ϭ {25, 40, shown by the solid lines in Fig. 12. The secondary ring 45, 67.5} km, {d1, d 2, d3, d 4} ϭ {2.5, 2.5, 2.5, 7.5} has the effect of ¯attening the tangential wind near r ϭ km, and {1, 2, 3, 4, 5} ϭ {47.65, 1.15, 9.65, 1.15, 42 km but no secondary wind maximum is present. The Ϫ0.35} ϫ 10Ϫ4 sϪ1, which simulates a large central retrograding vorticity waves along the central vortex 3910 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 57
FIG. 12. Azimuthal mean vorticity and tangential velocity for the experiment shown in Fig. 13 at the selected times t ϭ 0 (solid), 7.5 h (long dash), 9 h (medium dash), and 24 h (short dash).
edge (r ഠ 25 km) are embedded in relatively strong of the elliptical central vortex. During its evolution, the angular velocity of 23.5 ϫ 10Ϫ4 sϪ1 while the prograding central vortex takes on a variety of elliptical patterns waves along the inner edge of the ring (r ഠ 40 km) are with aspect ratios (major axis/minor axis) ranging from embedded in weaker angular velocity of 10.1 ϫ 10Ϫ4 1.3 to 1.8. Near the end of the simulation, the semimajor sϪ1. Near the outer edge of the ring (r ഠ 45 km), the (semiminor) axis is near 60 km (40 km) giving an aspect angular velocity is 8.8 ϫ 10Ϫ4 sϪ1, suppressing type 1 ratio near 1.5. Although the ellipticity of the central instability. Solving (A6) for this basic state reveals a vortex is a persistent feature, the tripole pattern becomes sole wavenumber-two instability whose growth rate has less easily identi®ed when t Ͼ 12 h as the vorticity of an e-folding time of 62 min. Results of this experiment the ring becomes increasingly mixed into the regions of are shown in Fig. 13. the satellites. At t ϭ 7.5 h, the wavenumber-two instability becomes The period of rotation of the tripole is approximately evident as the vortex and inner edge of the ring have 89 min. The linear theory of Kelvin (Lamb 1932, 230± been perturbed into ellipses. Asymmetries along the vor- 231) applied to a wavenumber-two asymmetry on a Ran- Ϫ1 tex are advected more quickly than those along the ring kine vortex with Vmax ϭ 58 m s at r ϭ 25 km predicts edge while the aspect ratios of the vortex and ring edge a period of rotation of 90 min. The good agreement move farther away from unity, so that at t ϭ 9h,the between Kelvin's theory and the model results suggest major vertices of the vortex have moved close enough that the period of rotation of a tripole can be predicted to the ring to begin stripping vorticity from it. The ®l- well if treated as a Kirchhoff vortex and is then in good aments of vorticity being stripped from the ring are then agreement with the results of Kuo et al. (1999). Polvani pulled across the moat into regions of stronger rotation and Carton (1990) calculated the rotation rate of a point- and are advected along the vortex edge. As the leading vortex tripole using the formula edges of the ®laments approach the downstream major vertices of the vortex, the differential rotation becomes 1 ⌫ ⍀ϭ ⌫ ϩ 2 , (20) greater and by t ϭ 10.5 h, the outer edges of the ®la- 2d2 1 2 ments have become trailing features. As the trailing edg- es approach the downstream major vertices, their inner where d is the distance from the centroid of the elliptical edges are again swept cyclonically along the edge of central vortex to the centroid of either satellite vortex the vortex. At t ϭ 12 h, this repetitive process has re- and ⌫1, ⌫ 2 are the circulations of the central and satellite sulted in two strips of vorticity, which originated from vortices. It is not clear how to accurately apply (20) to the ring edge, wound anticyclonically around two pools our numerical results since the calculation of ⌫ 2 requires of low vorticity, which originally composed the moat. quanti®cation of how much vorticity from the ring has The pools, or satellite vortices, lie along the minor axis been wound around the low vorticity of the moat. We 15 DECEMBER 2000 KOSSIN ET AL. 3911
FIG. 13. Vorticity contour plots for the type 2 instability experiment. Selected times are t ϭ 0, 7.5, 9, 10.5, 12, and 24 h. Successively darker shading denotes successively higher vorticity. 3912 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 57 can, however, approximate a range of values between not participate in the vorticity rearrangement and the two extreme cases, one in which the satellite vortices tripole pattern is more robust. contain only vorticity from the moat, and the other in The observational signature of a tripolar structure which the satellite vortices have ingested all of the vor- would likely be identi®ed by the presence of its weak ticity of the ring. At 24 h the centroid of each satellite satellite vortices, but the position where the satellites vortex is roughly 36 km from the centroid of the central would be expected to reside may inhibit such identi®- vortex and we can assume that the circulation of the cation. For example, airborne Doppler radar±derived central vortex has remained nearly ®xed so that ⌫1 ഠ ¯ows, as calculated by Reasor et al. (2000), can become 2.98 km2 sϪ1. If no vorticity from the ring were in- suspect at distances greater than 30±40 km from the gested by the satellite vortices, then their circulations hurricane center. In the experiment shown in Fig. 13, would be half the initial circulation of the moat, yielding the satellite vortices reside near r ϭ 40 km. The presence 2 Ϫ1 of such satellites in an actual hurricane could then be ⌫ 2 ഠ 0.06 km s . If all of the ring vorticity were ingested by the satellite vortices, then their circulations dif®cult to identify without some modi®cation to present would be half the sum of the initial circulations of the airborne radar-based methods. 2 Ϫ1 moat and ring, yielding ⌫ 2 ഠ 0.26 km s . Using these values in (20) we obtain the range of rotation 4. Concluding remarks periods 87 Շ P Շ 90 min, where P ϭ 2/⍀. It is clear that in this case, where ⌫ k ⌫ , there is little sensitivity The relative importance of conservative vorticity dy- 1 2 namics in a mature hurricane, compared with the non- to the choice of ⌫ 2 and the accuracy of (20) depends largely on the estimated value of d. conservative effects of friction and moist convection, is The evolution of the symmetric part of the ¯ow an open question. Ideally, the hurricane should be sim- (dashed lines in Fig. 12) shows a 26% reduction of the ulated using a progression of numerical models of vary- maximum vorticity in the ring during the initial stage ing complexity. Working from the nonhydrostatic prim- of the tripole formation between t ϭ 7.5 h and t ϭ 9 itive equations to the nondivergent barotropic model, h. This reduction smooths out the wind pro®le with the the implications of each simpli®cation can be measured initial ¯at spot replaced with a more uniform slope. At in terms of the question ``what physics remain?'' while progressing in the opposite sense, the question ``what t ϭ 24 h, the vorticity has become nearly monotonic physics are introduced?'' must be addressed. With more except for a small bump near r ϭ 50 km where trailing complex models, fundamental mechanisms may be ob- spirals are present, and the tangential wind maximum scured by the inclusion of additional but extraneous has decreased from its initial value of 58 to 52 m sϪ1. dynamics that often require parameterizations and typ- A remarkable aspect of type 2 instability is that a large ically must compromise, sometimes severely, numerical and intense central vortex can be dramatically perturbed resolution and thus dynamical accuracy. With simpler by apparently undramatic features of its near environ- models, there exists the danger that apparently dominant ment. For example, a thin ring of vorticity that is sta- mechanisms would be completely overshadowed by the bilized by its proximity to a central vortex can persist inclusion of additional physics. inde®nitely but is barely discernable in the wind ®eld. The present study was con®ned to the simple frame- Any subsequent strengthening or contraction of the ring work of the nondivergent barotropic model in an attempt can introduce type 2 instability, and the central vortex to capture the fundamental interaction between a hur- would be signi®cantly rearranged. ricane primary eyewall and surrounding regions of en- Unlike the case of an initial shielded vortex (section hanced vorticity in the context of asymmetric PV re- 3a), where wavenumber-two instabilities can saturate as distribution. A four-region model was used to simulate tripoles or dipole pairs, the emergence of a tripole in a monopolar central vortex surrounded by a region of our numerical simulations appears to be an ubiquitous low vorticity, or moat, which in turn was surrounded result of type 2 instability across a moat of reduced, but by an annular ring of enhanced vorticity. The moat was positive vorticity. Tripoles result from a variety of initial found to be a region of intense differential rotation and conditions that share the common feature of a depressed associated enstrophy cascade where cumulonimbus con- region of vorticity within the moat, with higher vorticity vection would have dif®culty persisting. This mecha- outside. The initial width of the annular ring of elevated nism offered an additional explanation beyond meso- vorticity is unimportant to the early development of a scale subsidence for the relative absence of deep con- tripole since the outer edge of the ring is dynamically vection in the moat. inactive. The width of the ring does, however, play a The four-region model provided a basic state that can role in the ¯ow evolution at later times. As seen in the support two instability types. In the ®rst instability type, numerical results of Fig. 14, for the case of an initially phase locking occured between vortex Rossby waves broad annular ring, the formation of a tripole is followed propagating along the inner and outer edges of the an- by a relaxation to a nearly steady solid body rotation nular ring. The central vortex was dynamically inactive that is absent in the previous experiment. In this case, but served to induce a differential rotation across the most of the vorticity within the interior of the ring does ring, which had the effect of stabilizing the ring by 15 DECEMBER 2000 KOSSIN ET AL. 3913
FIG. 14. Results of the numerical integration for the case of an initially wide annular ring. Mean pro®les of vorticity and tangential wind and vorticity contour plots for t ϭ 3.5 h and t ϭ 12 h. The outer edge of the ring does not participate in the mixing and the tripole rotates nearly as a solid body. Little change is observed in the mean pro®les after t ϭ 3.5 h.
opposing its self-induced differential rotation and in- Observed mean pro®les in Hurricane Gilbert, calculated hibiting phase locking between the ring edges. This sta- by Black and Willoughby (1992), suggested that the bilizing mechanism offered an explanation for the ob- presence of Gilbert's central vortex was indeed adequate served longevity of secondary eyewalls in hurricanes. to stabilize its secondary eyewall. In the case where the 3914 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 57 opposing differential rotation was insuf®cient to com- APPENDIX pletely stabilize the ring, the vorticity of the ring mixed to form a broader and weaker ring. During this process, Linear Stability Analysis of the the central vortex imposed a barrier to inward mixing Four-Region Model and inhibited rearrangement into a monopole. The ¯ow relaxed to a quasi-steady state, which satis®ed the Fjùrtoft suf®cient condition for stability while still sat- Consider a circular basic-state vortex whose angular isfying the Rayleigh necessary condition for instability, velocity (r) is a given function of radius r. Using and while maintaining a signi®cant secondary wind cylindrical coordinates (r, ), assume that the small- maximum. amplitude perturbations of the streamfunction, In the second instability type, the outer edge of the Ј(r, ,t), are governed by the linearized barotropic 2 )ٌ ЈϪץ/ץt ϩ ץ/ץ) annulus was rendered dynamically inactive by the vor- nondivergent vorticity equation 2 )(d/dr) ϭ 0, where (r) ϭ d(r )/rdr is theץЈ/rץ) tex-induced differential rotation, and interaction across ,ץЈ/rץthe moat between the inner edge of the annular ring and basic-state relative vorticity, (uЈ, Ј) ϭ (Ϫ -r) the perturbation radial and tangential compoץ/Јץ the outer edge of the central vortex took place. This 2 ϭٌ Ј theץuЈ/rץr ϪץrЈ)/r)ץ instability was most likely realized in azimuthal wave- nents of velocity, and number two and the subsequent mixing resulted in the perturbation vorticity. Searching for modal solutions of i(m Ϫt) formation of a tripole. This mechanism offered an ex- the form Ј(r, ,t) ϭ Ã (r)e , where m is the az- planation for the origin and persistence of elliptical eye- imuthal wavenumber and the complex frequency, we walls, but the existence and signi®cance of type 2 in- obtain the radial structure equation stabilities in tropical cyclones remains an open question. ddˆ d Acknowledgments. The authors would like to thank ( Ϫ m ) rr Ϫ m2ˆ ϩ mr ˆ ϭ 0. (A1) Rick Taft, Paul Reasor, Scott Fulton, William Gray, John []dr dr dr Knaff, Mark DeMaria, Frank Marks, Peter Dodge, Mel Nicholls, and Hung-Chi Kuo for many helpful com- A useful idealization of an annular region of elevated ments and discussions. This work was supported by NSF vorticity surrounding a central vortex is the piecewise Grant ATM-9729970 and by NOAA Grant constant four region model. In this model the axisym- NA67RJ0152 (Amendment 19). metric basic-state angular velocity (r) is given by
1 0 Յ r Յ r1, 1 Ϫ ( Ϫ )(r /r)2 r Յ r Յ r , (r) ϭ 2211 1 2 (A2) 22 2 3211Ϫ ( Ϫ )(r /r) Ϫ ( 322Ϫ )(r /r) r 2Յ r Յ r 3, 222 Ϫ(211Ϫ )(r /r) Ϫ ( 322Ϫ )(r /r) ϩ 33(r /r) r 3Յ r Ͻϱ,
and the corresponding basic-state relative vorticity by structed from different linear combinations of rm and rϪm in each region. A physically revealing approach is 110 Ͻ r Ͻ r , to write the general solution, valid in any of the four 2 d(r ) 21r Ͻ r Ͻ r 2, regions, as a linear combination of the basis functions (r) ϭϭ (A3) (m) Bj (r), de®ned by rdr 32r Ͻ r Ͻ r 3, 0 r Ͻ r Ͻϱ, 3 (r/r )0m Յ r Յ r , B(m)(r) ϭ jj(A4) j m where r1, r 2, r3 are speci®ed radii and 1, 2, 3 are Ά(rjj/r) r Յ r Ͻϱ, speci®ed vorticity levels. Restricting study to the class of perturbations whose for j ϭ 1, 2, 3. The solution forà (r) is then à (r) ϭ (m)(m)(m) disturbance vorticity arises solely through radial dis- ⌿1BBB123(r) ϩ⌿2 (r) ϩ⌿3 (r), where ⌿1, ⌿2, and (m) placement of the basic-state vorticity, then the pertur- ⌿3 are complex constants. SincedB1 /dr is discontin- bation vorticity vanishes everywhere except near the uous at r ϭ r1, the solution associated with the constant edges of the constant vorticity regions, that is, (A1) ⌿1 has vorticity anomalies concentrated at r ϭ r1 and 2 reduces to (rd/dr)(rdˆ /dr ) Ϫ m ˆ ϭ 0 for r ± r1, r2, the corresponding streamfunction decays away in both r3. The general solution of this equation in the four directions from r ϭ r1. Similarly, the solutions asso- regions separated by the radii r1, r 2, r3 can be con- ciated with the constants ⌿2 and ⌿3 have vorticity 15 DECEMBER 2000 KOSSIN ET AL. 3915
FIG. A1. Isolines of the maximum value of the dimensionless growth rate i/3, computed from (A7), as a function of
r1/r2 and r2/r3 for the case 1/3 ϭ 5.7 and for azimuthal wavenumbers up to m ϭ 16. Type 1 instability (for m ϭ 3, 4, ...,16)occurs on the left side of the diagram and type 2 instability (for m ϭ 2,3,...,16)occurs on the right side. The azimuthal wavenumber associated with the most unstable mode is indicated by the alternating gray scales. The white region
is stable, and the isolines are i/3 ϭ 0.01, 0.03, 0.05, ....
r ϩ⑀ anomalies concentrated at r ϭ r 2 and r ϭ r3, respec- dˆ j tively. lim ( Ϫ mjj)r ϩ ( jϩ1 Ϫ jj)mˆ (r ) → ⑀ 0 dr r In order to relate ⌿ , ⌿ , ⌿ , we integrate (A1) over []jϪ⑀ 1 2 3 the narrow radial intervals centered at r1, r 2, r3 to obtain the jump (pressure continuity) conditions ϭ 0, (A5) 3916 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 57
where j ϭ (rj) and j ϭ 1, 2, 3. Substituting our general solution into the jump conditions (A5) yields the matrix eigenvalue problem 11 1 m ϩ ( Ϫ )( Ϫ )(r /r )(m Ϫ )(r /r )m 12122 21122113 2 ⌿⌿ 111 11 ( Ϫ )(r /r )m m ϩ ( Ϫ )( Ϫ )(r /r )m⌿ϭ ⌿ . (A6) 2223212 2 32 32232 2 ⌿⌿ 11 133 Ϫ (r /r )m Ϫ (r /r )m m Ϫ 22313 323 3 2 3
1 Noting that for the basic state given by (A2), 1 ϭ 2 1, Now consider the special case obtained by 112 2 ϭ 22[ 2 Ϫ ( 2 Ϫ 1)(r1/r 2) ] and 3 ϭ [3 Ϫ ( 2 Ϫ assuming 2 ϭ 0. Then, dividing the ®rst row of 2 2 11 1)(r1/r3) Ϫ (3 Ϫ 2)(r 2/r3) ], we can solve the ei- (A6) by 221 and the second and third rows by 3 ,we genvalue problem (A6) once we have speci®ed the pa- obtain rameters m, r1, r 2, r3, 1, 2, 3.
ΗΗϪ1 rrmm 1 Ϫ m ϩ 2 11 1 33 rr 2 3 ΗΗm 2 m r1112 r r ΗΗ1 ϩ m Ϫ 2 ϭ 0. (A7) r23233 r r rrmm rr22 ΗΗ121 Ϫ m 1 ϩϪϩ 1122 rr rr ΗΗ33 [] 3333
For given r1/r 2, r 2/r3, and 1/3, the determinant (A7) k r3 Ϫ r 2, where instabilities across the ring strongly yields three values of the dimensionless frequency /3. dominate instabilities across the moat. The region of Choosing the most unstable of these three roots, Fig. Fig. A1 that is pertinent in this limit is restricted to the
A1 shows isolines of the imaginary part of /3 as a upper-left corner and, in this case, instabilities that serve function of r1/r 2 and r 2/r3 for the case 1/3 ϭ 5.7. For as a precursor to tripole formation are suppressed. Car- this value of 1/3, there are two distinct types of in- ton and Legras considered the special case of the four- stability. Type 1 occurs for azimuthal wavenumbers m region model applied to shielded monopoles. They pre-
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