6.2 VORTEX INTERACTIONS AND THE BAROTROPIC ASPECTS OF CONCENTRIC EYEWALL FORMATION H.-C. Kuo ∗ National Taiwan University, Taipei, Taiwan W. H. Schubert Colorado State University, Fort Collins, Colorado 1. INTRODUCTION vorticity of alternate signs, with the whole configuration steadily rotating in the same sense The general interaction of two vorticity as the vorticity of the elliptically shaped central patches with the same vorticity, but with different core (Carton et al. 1989, Polvani and Carton horizontal areas, has been described by 1990, Carton and Legras 1994, Kloosterziel and Dritschel and Waugh (1992). Based on a Carnevale 1999). Examples of elliptical eyes that quantification of the final to initial circulation of resemble a tripole vortex structure were reported each vortex, the interactions can be classified by Kuo et al. (1999) for the case of Typhoon into five different types: elastic interaction, partial Herb (1996), and by Reasor et al. (2000) for the straining out, complete straining out, partial case of Hurricane Olivia (1994). merger, and complete merger. There are many Based on the arguments of Okubo (1970) tropical cyclone observations that resemble and Weiss (1991), Rozoff et al. (2006, hereafter these binary vortex interaction regimes (e.g., see R06) examined the rapid filamentation zones Kuo et al. 2000, Prieto et al. 2003). In the that form in intense tropical cyclones. They complete straining out regime the smaller vortex pointed out that the strain-dominated flow region is drawn out into thin filaments of vorticity just outside the radius of maximum wind of the surrounding the larger vortex, with no core vortex can contribute significantly to the incorporation of fluid into the large vortex. This moat dynamics. Namely, the strong differential regime resembles the concentric vorticity rotation outside the radius of maximum wind of structure of tropical cyclones, except the vorticity the core vortex may also contribute to the filaments are probably too thin to be called a formation and maintenance of the moat. They concentric eyewall. Observations of Typhoon also note that one way to produce a concentric Lekima (Kuo et al. 2004, hereafter K04) indicate vorticity structure is through the interaction that it had a large area of convection with weak between a strong core vortex and a background cyclonic vorticity outside the core vortex and that turbulent vorticity field. The vorticity halo this weak vorticity wrapped around the inner produced in their experiment, however, has only eyewall on a time scale of 12 hours. This half the magnitude of that produced in typical scenario can be idealized as the binary binary vortex interactions. The effect of vorticity interaction of a small and strong vortex (the skirts on binary vortex interaction and the effect tropical cyclone core) with a large and weak of turbulent background vorticity on the vortex (the vorticity induced by moist convection formation of concentric vorticity structures are outside the core vortex). This type of binary the focal points of this paper. The paper extends interaction was not studied by Dritschel and the work of K04 and R06 by including an Waugh, since their vortices were assumed to extended vorticity gradient outside the radius of have the same strength and their larger vortex maximum wind in the binary vortex interaction as was always the “victor.” With the introduction of a well as by exploring the concentric eyewall “vorticity strength parameter” into the binary formation in a turbulent background vorticity with interaction problem with Rankine vortices, K04 various characteristic spatial scales. Passive added a third dimension to the Dritschel-Waugh microwave observations of secondary eyewall parameter space and two new types of resulting formation are presented in section 2. Section 3 interaction: concentric vorticity structure and describes the solution method and the model tripole vorticity structure. A tripole is a linear parameters. Section 4 gives the model results. arrangement of three regions of distributed The summary and concluding remarks are given ______________________________________ in section 5. ∗ Corresponding author address: H.-C. Kuo, 2. PASSIVE MICROWAVE OBSERVATIONS Dept. of Atmospheric Science, National Taiwan OF SECONDARY EYEWALL FORMATION University, Taipei, Taiwan; e-mail: kuo@lanczos. as.ntu.edu.tw From a study of passive microwave data from 1997 to 2002, Hawkins and Helveston separation distance to either decrease or (2004) concluded that concentric eyewalls exist increase, depending on the direction of the in a much higher percentage of tropical cyclones vorticity gradient. A vortex with a negative radial than previously estimated from visible and gradient of vorticity will be more merger-prone. infrared satellite sensors. Although based on a Moreover, the slower decrease of angular small sample, their results suggest that velocity associated with the extended vorticity approximately 40% of the Atlantic, 60% of the field should slow the filamentation process and Eastern Pacific, and 80% of the Western Pacific thus the moat formation. intense storms (maximum wind ≧ 120 knots) Mallen et al. (2005) examined the swirling have concentric eyewalls. Figure 1 shows 3 wind structure of tropical cyclones by utilizing western Pacific storms with concentric eyewalls, flight-level observations collected from Atlantic as viewed with passive microwave sensors. The and eastern Pacific storms during 1977—1999. data are from the Naval Research Laboratory Their results indicate that tropical cyclone Marine Meteorology Division in Monterey, CA structure is characterized by a relatively slow (NRL-MRY) (Hawkins et al., 2001). In each case, tangential wind decrease outside the radius of the time period shown is approximately 12 hours. maximum wind and a corresponding skirt of Initially the deep convection, indicated by the significant cyclonic relative vorticity. The Rankine brown color, tends to be quite asymmetric with vortices used in K04 have zero vorticity outside respect to the core and to possess random the radius of maximum wind and hence a rapid turbulent features. decrease of angular velocity with radius outside In Typhoons Imbudo, Dujuan, and Maemi, a the core. large area of convection outside the core vortex appears to wrap around the inner eyewall to 3. NONDIVERGENT BAROTROPIC MODEL form a concentric eyewall. The initial separation distance of the outer deep convection region The basic dynamics considered here is from the vortex core varies from case to case. two-dimensional, nondivergent, barotropic with For example, the outer deep convection almost ordinary diffusion in a double periodic domain. touches the vortex core in Typhoon Dujuan, The discretization of the model is based on the while the outer deep convection is some 100 km Fourier pseudospectral method, with 512 by 512 away from the vortex core in Typhoon Maemi. equally spaced collocation points on a 300 km by Figure 1 suggests that a symmetric structure can 300 km domain. The code was run with a evolve from asymmetric convection on a time dealiased calculation of quadratic nonlinear scale of approximately 12 hours. The microwave terms with 170 Fourier modes in each direction. imagery also illustrates moats of different sizes Time differencing was via the fourth-order in the concentric eyewall cases. Even though the Runge-Kutta method with a 3 second time step. initial separation distances are different in The diffusion coefficient, unless otherwise Typhoons Imbudo, Dujuan, and Maemi, the specified, was chosen to be ν = 6.5 m2s-1. For concentric eyewalls possess a value of the 300 km by 300 km domain this value of ν approximately unity for the ratio of moat width to gives an e-1 damping time of 3.37 hours for all core radius. In contrast, the largest moat is found modes having total wave number 170, and a in Typhoon Winnie, with its outer eyewall at a damping time of 13.5 hours for modes having radius of 275 km (Zhang et al. 2005), which total wave number 85. Some of the experiments yields a value of 6 for the ratio of moat width to were repeated at increased resolution and/or core radius. with a larger domain size. From these The explanation of an interaction over such experiments we conclude that the results shown a large distance requires an extension of the here are insensitive to the domain size and to previous binary vortex interaction results of K04, the resolution employed. Obviously, the use of who studied only vortices with sharp edges (i.e. so simple a model precludes the simulation of unskirted vortices). It should be noted that the complete secondary eyewall cycle, but it vorticity skirts play a role in other aspects of allows for some simple numerical experiments tropical cyclone dynamics. For example, concerning the initial organizational processes DeMaria and Chan (1984) argued that mergers involved in secondary eyewall formation. in binary vortex interaction can also occur due to We consider an initial condition consisting of vortex propagation on the outer vorticity two distinct vortices—a strong, skirted, core gradients associated with each vortex. The vortex and a weaker, larger, unskirted interaction of the tangential wind field with the companion. The initial condition contains the six outer vorticity field of the companion vortex adds parameters ζ1, ζ2, R1, R2, d, α, where the ζ1, ζ2 a component to the motion that can cause the are the vorticity field, R1, R2 the measure of vortex size, d the distance between the center, Figure 1. The wind profiles clearly show a and α the skirt parameter. The initial vorticity secondary maximum in the tangential wind field field is given by contracting with time. The secondary wind maximum increases from the initial 25 ms-1 to 40 1, if 0 ≤ r1 ≤ 0.65, -1 2 3 ms in 12 hours. The time and spatial scales of ζ (x, y,0) = ζ 1 c0 + c1r1 + c2r1 + c3r1 , if 0.65 ≤ r1 ≤ 0.81, 1 −α −1 the secondary wind maximum contraction in 2 (1−α )r1 , if 0.81 ≤ r1 ≤ ∞, (1) 30 1 Figure 3 are in general agreement with the 1- exp[]- exp() , if 0 ≤ r2 ≤ 1, r2 r2 −1 + ζ 2 observations in Hurricane Gilbert (Black and 0, if 1 ≤ r ≤ ∞, 2 Willoughby 1992).