the primary eyewall into a symmetric band that encircled the eyewalls. In a concentric eyewall, both the deep convection and the potential vor-ticity are enhanced. Thus, in terms of potential vorticity dynamics, the formation of a concentric eyewall would appear to have two important aspects: (i) the organization of an existing asymmetric potential vorticity distribution into a symmetric potential vorticity distribution with a moat; (ii) the diabatic enhancement of potential vorticity during the organizational process. Although both aspects are probably important in most examples, it is useful to isolate the first aspect through the study of highly idealized dynamical models. Namely, one way to produce the concentric eyewall structure is through a binary vortex interaction such as those reported in K04 and K06.
This article reviews some important points from K04 and K06 in the dynamics of tropical cyclone concentric eyewall formation. We argue that the organization of the asymmetric convection into a symmetric concentric eyewall can be accomplished through pure advective processes of vorticity dynamics. We address the dynamics of why there is a wide range of radii for concentric eyewalls, and the roles of the moat and the turbulent vorticity scale in the formation of concentric eyewall structure. Section 2 describes the model and the solution method. The physical parameters are discussed in Sec. 3. The numerical results of binary vortex interactions and the results of concentric eyewall formation solely from a turbulent background are presented in Sec. 4. Section 4 also contains a discussion on the two-dimensional turbulence characteristic of the model as well as the regime phase space diagrams. Concluding remarks are made in Sec. 5.
The basic dynamics considered is two-dimensional nondivergent barotropic with ordinary diffusion, i.e. DQ/Dt = vV2Z, where D/Dt = d/dt + u(d/dx) + v(d/dy). Expressing the velocity components in terms of the stream function by u = -d—/dy and v = d—/dx, we can write the nondivergent barotropic model as where
is the invertibility principle and d(, )/d(x,y) is the Jacobian operator. The diffusion term on the right hand side of (1) controls the spectral blocking associated with the enstrophy cascade to higher wave numbers. As discussed in K04 and Prieto et al. (2001), we have avoided the use of hyperviscosity (higher iterations of the Laplacian operator on the right hand side of the vorticity equation) because of the unrealistic oscillations it can cause in the vorticity field. We perform calculations on a doubly periodic square domain. The Fourier pseudospectral method is employed for the discretization, with 512 x 512 equally spaced collocation points on a 300 km x 300 km domain for the vortex interaction experiments. The code was run with a dealiased calculation of quadratic nonlinear terms with 170 x 170 Fourier modes. The fourth-order Runge-Kutta method with a 3s time step is used for the time differencing. The diffusion coefficient, unless otherwise specified, was chosen to be v = 6.5 m2s-1. For the 300 km x 300 km domain, this value of v gives an e-1 damping time of 3.37 h for all modes having total wave number 170, and a damping time of 13.5 h for modes having total wave number 85. Some of the experiments were repeated at increased resolution and/or with a larger domain size. From these experiments, we conclude that the results shown here are insensitive to the domain size and to the resolution employed. The use of such a simple model obviously precludes the simulation of the complete secondary eyewall cycle, but it allows for some simple numerical experiments concerning the initial symmetrization dynamics involved in secondary eyewall formation.
Our experiments can be viewed as an extension of those of Dritschel and Waugh (1992; hereafter DW92) and Dritschel (1995). DW92 described the general interaction of two barotropic vortices with equal vorticity but different sizes. They performed experiments on the /-plane by varying the ratio of the vortex radii and the distance between the edges of the vortices normalized by the radius of the larger vortex. The resulting structures can be classified into elastic interaction, merger, and straining-out regimes. The elastic interaction involves distortion to the vortices in a mutual cyclonic rotation. The merger regime involve part of the smaller vortex being removed, and some of it being incorporated into the larger vortex. In the complete straining-out regime, a thin region of filamented vorticity bands surrounding the central vortex with no incorporation into the central vortex. This appeared to resemble a concentric vorticity structure with a moat. However, the outer bands which result from the smaller vortex are much too thin to be identified with that observed in the outer eyewall of a tropical cyclone.
There are four parameters used in the DW92 study: the vorticity Z, the vortex sizes R1 and R2, and the separation distance between the edges of the two vortices A. With the specification of R1 and Z, the end states of binary vortex interaction are summarized by the remaining degrees of freedom: R2/R1 and A/Ri. The interaction of a small and strong vortex with a large and weak vortex was not studied by DW92 as their vortices are of the same strength (i.e. the same Z), and their larger vortex was always the "victor" and the smaller vortex was the one being partially or totally destroyed. Observations of Typhoons Lekima and Imbudo, as seen in Fig. 1, indicate that there may be a huge area of convection with weak cyclonic vorticity outside the core vortex that wraps around the inner eyewall, rather than the other way around. An extension of the complete straining-out regime to include a finite-width outer band is needed to explain the interaction of a small and strong vortex (representing the tropical cyclone core) with a large and weak vortex (representing the vorticity induced by the moist convection outside the central vortex of a tropical cyclone).
We have introduced the sharp-edged vorti-city patches Z1 and Z2 to replace the DW92 Z parameter in K04. With the specification of the vorticity of the companion vortex Z2 and the size of the core vortex R1 = 10 km, the number of parameters is reduced to three, which we take to be the vorticity strength ratio 7 = Z1/Z2, the vortex radius ratio r = R1 /R2, and the di-mensionless gap A/R1. Using aircraft flight-level data, Mallen et al. (2005) demonstrated that tropical cyclones are often characterized by a relatively slow decrease of tangential wind outside the radius of maximum wind, and hence by a corresponding cyclonic vorticity skirt. To include the skirt effect, we have set the radial profile of vorticity of constant vorticity Z1 in the inner region and of the skirted structure ^Ci(l ~~ a)(r/R1 )-a-1 in the outer region. The constant R1 is a measure of the core vortex size, and a is the nondimensional skirt parameter. In the vor-ticity skirt region, the azimuthal wind behaves as r-a. The a = 1 profile is the Rankine vortex profile, which has zero vorticity gradient and rapid decrease of angular velocity with radius outside the core. Aircraft observations of the azi-muthal winds in hurricanes (e.g. Shea and Gray, 1973; Mallen et al., 2005) suggest that a reasonable range for the skirt parameter is 0.5 < a < 1. In contrast to the strong core vortex, the companion vortex is not skirted. The omission of a vorticity skirt on the companion vortex is based on our experience that the vorticity skirt on the "victorious" core vortex is more important. In addition, it is desirable to keep the number of parameters to a manageable level.
Vorticity skirts play a role in several aspects of tropical cyclone dynamics. For example, vortex mergers can occur owing to vortex propagation on the outer vorticity gradients associated
Vortex Interactions and Typhoon Concentric Eyewall Formation Table 1. Summary of the experimental parameters.
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