R2 A

C2 ' R2 ' Rl with each vortex (DeMaria and Chan, 1984). In addition, the radial vorticity gradient associated with the vorticity skirt provides a means for the radial propagation of vortex Rossby waves (Montgomery and Kallenbach, 1997; Balmforth et al, 2001). Finally, skirted vortices can be more resilient to the destructive effects of large-scale vertical wind shear (Reasor et al., 2004). As far as the concentric eyewall fomation dynamics are concerned, the skirt on the core vortex provides a shielding effect which may lead to the formation of outer bands further away. In addition, the slower radial decrease of angular velocity associated with the vorticity skirt lengthens the filamentation time and slows moat formation.

In summary, we have used the skirt parameter a, the vorticity strength ratio y = Z1/Z2, the vortex radius ratio r = R1 /R2, and the dimensionless gap A/R1 to classify our model end states. Table 1 gives a summary of the parameters, the specified conditions, and dimension-less parameters that were used for the end state classification for the DW92, K04, and K06 papers. Figure 2 shows the initial schematic configuration of two circular vortices in DW92, K04, and K06. With the introduction of the vortici-ty strength ratio and the skirt parameter a into the binary vortex interaction problem, we have

Figure 2. Initial configuration of two circular vortices with radii R\ and R2, vorticity £1 and Z2 , the skirt parameter a and the gap A. The R1 > R2 and Zi = Z2 = Z in DW 92, and Ri < R2 and Zi > Z2 in K04 and K06.

added a new dimension to the Dritschel-Waugh vortex interaction scheme which provides a proper concentric vorticity structure of various sizes as well as the tripole vortex structure.

4. Numerical Results

4.1. Binary vortex interaction

Figure 3 shows the vorticity field in the binary vortex experiments with r= 1/4 for the (a) concentric case (7 = 8, A/R1 =0, a = 1.0), (b) merger case (7 = 8, A/R1 = 0, a = 0.5), (c) elastic interaction (7 = 6, A/R1 = 5, a = 1.0), and (d) concentric case (7 = 6, A/R1 = 5, a = 0.5). The figure suggests the formation of concentric vorticity structures for both skirted a = 0.5 and un-skirted (a = 1.0) vortices. The Rankine core vortex (a = 1.0) produces the concentric structure with the zero initial separation distance. On the other hand, the skirted core vortex

(a = 0.5) produces a merger with the same zero initial separation distance. With a separation distance that is five times the core vortex radius, the Rankine vortex produces inelastic interaction, and the a = 0.5 vortex produces concentric structure. Thus, Fig. 3 suggests that a skirted core vortex of sufficient strength can form a concentric vorticity structure at a larger radius than what is allowed by an unskirted core vortex. This may explain the wide range of radii for concentric eyewalls in observations.

Even though the change of sign of the vor-ticity gradient across the outer band satisfies the Rayleigh necessary condition for barotropic stability, the band is stabilized by the Fj0rtoft r=1/4

200km

Figure 3. Vorticity fields in the binary vortex experiments with the r = 1/4 parameters for the (a) concentric case (Y = 8, A/Ri = 0, a = 1.0), (b) merger case (7 = 8, A/Ri = 0, a = 0.5), (c) elastic interaction (7 = 6, A/Ri = 5, a = 1.0), and (d) concentric case (7 = 6, A/Ri = 5, a = 0.5).

200km

Figure 3. Vorticity fields in the binary vortex experiments with the r = 1/4 parameters for the (a) concentric case (Y = 8, A/Ri = 0, a = 1.0), (b) merger case (7 = 8, A/Ri = 0, a = 0.5), (c) elastic interaction (7 = 6, A/Ri = 5, a = 1.0), and (d) concentric case (7 = 6, A/Ri = 5, a = 0.5).

sufficient condition for stability. Namely, the strong inner vortex causes the wind to be stronger at the inner edge than at the outer edge, allowing the vorticity band and therefore the concentric structure to be sustained. A similar mechanism was discussed by Dritschel (1989) and Polvani and Plumb (1992), who showed how thin filaments can be stabilized by the flow field of the main vortex. They argued that the filament is linearly stable and appears circular in the presence of sufficiently strong "adverse shear." The adverse shear is an externally controlled parameter with the opposite sense to that produced by the filament's vorticity alone. In general, the inner vortex must also possess high vorticity not only to be maintained against any deformation field induced by the outer vortices, but also to maintain a smaller enstrophy cascade and to resist the merger process into a monopole. A detailed discussion can be found in K04.

Figure 4 shows the tangential wind speed from the aircraft observations of Hurricane Gilbert (BW92), and from a K04 model concentric eyewall simulation, along radial arms emanating from the core vortex center. These observations show the contraction of the outer tangential wind maximum from a distance of 90 km from the storm center to 60 km in approximately 12 hours. The core vortex intensity remained approximately the same during the contraction of the outer tangential wind maximum. Wind profiles from the model also clearly show the contraction of secondary maxima in the tangential wind field. It appears that the contraction of the secondary wind maxima in the model are in general agreement with observations of Hurricane Gilbert. The contraction mechanism for outer bands is often argued to be a balanced response to an axisymmetric ring of convective heating (Shapiro and Willoughby, 1982). The results shown in Fig. 4 suggest that the nonlinear advective dynamics involved in the straining-out of a large, weak vortex into a concentric vorticity band can also result in contraction of the secondary wind maximum. No moist convection is involved in this process. However, moist convection could substantially enhance the strength of such a band.

4.2. Two-dimensional turbulence on the f-plane

We now address the two-dimensional turbulence characteristics that are relevant to our tripole and concentric eyewall formation dynamics. We restrict our attention to /-plane dynamics, so the Rhines scale dynamics will not be discussed. Three important integral properties from our model equations (1) and (2) during the numerical simulations are the energy E = ff ^Vip ■ Vip dxdy, the enstrophy Z = ff 2 dxdy, and the palinstrophy P = ff ^V( ■ V( dxdy. The pa-linstrophy is a measure of the overall vorticity gradient in the domain. If there is no flux across the boundary, as can be shown from (1) and (2), these three integral properties are related by dE ~~dt dZ ~dt

and the palinstrophy by dp rrfd^x)

V Z dxdy

0 0

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