2656 MONTHLY WEATHER REVIEW VOLUME 131 A Classi®cation of Binary Tropical Cyclone±Like Vortex Interactions* RICARDO PRIETO Centro de Ciencias de la AtmoÂsfera, Universidad Nacional AutoÂnoma de MeÂxico, Mexico City, Mexico BRIAN D. MCNOLDY Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado SCOTT R. FULTON Department of Mathematics and Computer Science, Clarkson University, Potsdam, New York WAYNE H. SCHUBERT Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado (Manuscript received 3 December 2002, in ®nal form 14 April 2003) ABSTRACT The interaction between two tropical cyclones with different core vorticities and different sizes is studied with the aid of a nondivergent barotropic model, on both the f plane and the sphere. A classi®cation of a wide range of cases is presented, using the Dritschel±Waugh scheme, which subdivides vortex interactions into ®ve types: elastic interaction, partial straining out, complete straining out, partial merger, and complete merger. The type of interaction for a vortex pair on the f plane, and the same pair on the sphere, was the same for 77 out of 80 cases studied. The primary difference between the results on the f plane and those on the sphere is that the vorticity centroid of the pair is ®xed on the f plane but can drift a considerable distance poleward and westward on the sphere. In the spherical case, the interaction between the cyclone pair and the associated b- induced cyclonic and anticyclonic circulations can play an important role. The ``partial merger'' regime is studied in detail. In this regime the interaction between vortices can lead to episodic exchanges of vorticity, with both vortices surviving and entering a stage of continued but weaker interaction. With the aid of passive tracers, it is found that the exchange of vorticity is restricted to the vortex periphery even when the vorticity ®eld within each vortex is ¯at, so that the vortex core is the last region to be eroded. It is hypothesized that some observed interacting tropical cyclones actually do undergo this partial- merger process. 1. Introduction ward but moving 50% faster than Gil. On the morning An interesting tropical cyclone binary interaction oc- of 6 September, Gil had reached hurricane strength, and, curred in the east Paci®c during September 2001. The at about the same time, Henriette had reached tropical event began when Tropical Depression 8 formed on the storm strength. Gil was a relatively compact, intense 21 morning of 4 September. Just 12 h later, it was upgraded circulation (42 m s peak intensity), while Henriette 21 to Tropical Storm Gil and was tracking westward. Lag- was a broader, weaker circulation (22 m s peak in- ging just a few hours and 1500 km behind Gil was tensity). Figure 1a is a GOES-10 visible image of these Tropical Depression 9, which would later become Trop- two storms on 6 September 2001 at 2100 UTC. (GOES- ical Storm Henriette. Henriette was also tracking west- 10 visible loops for 6±9 September are available in a supplement found online at http://ams.allenpress.com/ amsonline/?request5index-html.) * Supplemental information related to this paper is available online By the morning of 8 September, the separation be- (DOI:10.1175/MWR2610Supl1). For current information see http:// dx.doi.org/10.1175/MWR2610Supl1. tween the two vortices had decreased to only 400 km, and signs of vortex interaction began to appear (strong shear aloft and a low-level horizontal convergence Corresponding author address: Dr. Ricardo Prieto, Instituto Mex- band). Also around this time, Henriette had lost icano de Tecnologia del Agua, Paseo Cuauhnahuac 8532, Jiutepec, Mor. 62550, Mexico. strength and was no longer classi®ed as a tropical de- E-mail: [email protected] pression. On the afternoon of 9 September, the centers q 2003 American Meteorological Society NOVEMBER 2003 PRIETO ET AL. 2657 FIG.1.(a)GOES-10 visible image of Hurricane Gil and Tropical Storm Henriette at 2100 UTC 6 Sep 2001. Henriette is to the northeast. Latitude and longitude lines are 58 apart, and Gil is at 158N, 1288W. (b) Tracks of Gil and Henriette based on NHC best-track positions and GOES-10 visible imagery, shown with their geographic centroid (dashed line). 2658 MONTHLY WEATHER REVIEW VOLUME 131 of Gil and Henriette had come to within 85 km of each TABLE 1. Grid parameters at t 5 0: grid number (l), number of grid points in x and y (n ,n), grid origin (x , y ), grid spacing in x other, already having rotated 5408 about their centroid. x y 0 0 and y (h ), and time step (Dt ). By this time, Gil was also no longer a tropical de- l l pression. Finally, by the morning of 10 September, the lnx ny x0 (km) y0 (km) hl (km) Dtl (s) vortices had merged, with the more compact, intense 1 512 512 25223.63 24613.87 20.4048 450.00 one (remnants of Gil) victorious. The storm tracks, 2 512 512 22611.82 22614.19 10.2024 225.00 based on National Hurricane Center (NHC) best-track 3 512 512 21305.91 21308.29 5.1012 112.50 positions and GOES-10 visible imagery, are shown in 4 512 512 2652.95 2655.33 2.5506 56.25 Fig. 1b. This ®gure shows the geographic centroid; the vorticity centroid could be closer to the stronger storm but cannot be computed without the detailed vorticity ®eld. parameter on the interaction between two tropical cy- The binary vortex interaction is also known as the clones, concluding that the development of ``b gyres'' ``Fujiwhara effect'' in honor of the pioneering work can have an important impact on the vortex interaction. of Fujiwhara (1921, 1923, 1931), who performed a The b-gyres are the result of the meridional advection series of laboratory experiments on the interactions of planetary vorticity around a vortex, which induces between pairs of vortices in a water tank. The inter- an asymmetric circulation capable of driving the vortex. action of Gil and Henriette is one example of the many b-gyres of suf®cient intensity can extend the distance cases of binary tropical cyclone interactions that have at which one vortex is able to capture the other, or they been documented in the literature (e.g., see Larson can produce the escape mode discussed by Lander and 1975; Chang 1983; Lander and Holland 1993; Kuo et Holland (1993). al. 2000; Wu et al. 2003). In an early observational From the theoretical point of view, Melander et al. study of the interaction of binary tropical cyclones in (1986) developed a vortex interaction model by con- the western North Paci®c, Brand (1970) found that a sidering the evolution of ®nite-area, uniform-vorticity de®nitive mutual interaction, in the form of a cyclonic regions in an unbounded inviscid ¯uid. Their analysis orbit of the storms about their centroid, occurred when of the two-dimensional Euler equations results in a the cyclone centers came within 1300 km, and that system of ordinary differential equations governing mutual attraction occurred when the cyclone centers the physical-space moments of the individual regions. came within 750 km. Of course, these should be re- Truncation of this in®nite system yields an nth-order garded as rough forecasting rules since tropical cy- moment model. In the second-order model each re- clones vary widely in their sizes and strengths. In the gion is assumed to be elliptical, and the equations of western North Paci®c the frequency of such binary motion conserve local area, global centroid, total an- interactions is approximately 1.5 per year. The fre- gular impulse, and global excess energy. The model quency of actual merger is lower. forms a Hamiltonian system, with the excess energy The interaction of two isolated vortices has been as the Hamiltonian. This model can crudely describe examined in detail in a numerical study by Dritschel the collapse of two vortex centers. However, as the and Waugh (1992) in which the vortices are patches vortex centers approach during merger, the model be- with equal vorticity, and the radius ratio and separation comes increasingly invalid because it assumes that an distance are experimental parameters. The calcula- elliptical shape is maintained throughout the process. tions, performed on the f plane using the contour sur- Another theory of vortex merger has been presented gery algorithm (Dritschel 1988), show that the inter- by Lansky et al. (1997). In their theory, a point vortex action of unequal vortices is much richer than that of and an extended vortex of nearly circular cross section equal vortices. The interactions can be classi®ed into interact. The general behavior of this system involves ®ve different regimes, based upon a quanti®cation of the rotation of the point vortex around the extended the ®nal to initial circulation of each vortex: (i) elastic vortex, together with ®nite-amplitude oscillations in interaction, (ii) partial straining out, (iii) complete the radial direction. The radial motion of the point straining out, (iv) partial merger, and (v) complete vortex is trapped inside a critical layer of characteristic merger. These interactions may result in the total de- width. Lansky et al. (1997) found that if the total cir- struction of one of the vortices (iii and v), the partial culation of the point vortex divided by the total cir- destruction of one vortex (ii and iv), or an interaction culation of the extended vortex exceeds a critical value, in which both vortices survive with their initial cir- there is an overlap of neighboring critical layers around culation (i). These results also imply that the interac- the extended vortex, having as a consequence the cas- tion of same-sign vortices is an essential mechanism cade of the point vortex through a sequence of reso- for vortex growth, as well as an important mechanism nances as it spirals in and merges with the extended for the production of small vortices and for the de- vortex.