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The shallow-water-equation model used in this study includes the following equations in the Cartesian coordinates:
where u, v, V denote the zonal, meridional wind components, and wind vector, respectively; f = 2
${ \Omega }$ sin$ \varphi $ with$ \varphi $ being the latitude. The model domain has 3203 × 2403 grid points with a grid spacing of 3 km. Results within an area of 601 × 601 grid points centered at the vortex center are saved for analysis. Rigid lateral boundary conditions are used at the north and south boundaries, and cyclic conditions are used at the east and west boundaries. The combined scheme of Euler forward- and Matsuno time differencing is adopted for the model time integration with a time step of 9 s, while the simulated data are saved at 15-minute intervals. Note that our attention is given to the internal TC dynamics and the beta effect. Thus, we assume a quiescent environment.Observations suggest that intensifying TCs generally have very thin, high-PV annuli accompanied by nearly irrotational flow at the center (Schubert et al., 1999; Kossin and Eastin, 2001; Kossin and Schubert, 2001). To meet the two necessary conditions for barotropic instability presented in section 2, the model is initialized with an axisymmetric annulus of elevated PV (i.e., a PV ring), which has a radial distribution given as
where
${{q}}_{\text{r0}}$ is the maximum PV of the PV ring; rr is the radial distance to the TC center; ro is the radius at which the annulus is centered; l is the width of the annulus; the Coriolis parameter f is set at the latitude of 15°N; and$\overline{H}$ is the basic-state geopotential height. This radial profile indicates a nearly zero PV in the eye region with an increase in PV with increasing radius until a threshold radius given by ro – l/2, then remains nearly constant before decreasing outward from another threshold radius given by ro + l/2. Since PV and the inertial parameter are always positive in this case, such a structure satisfies the Charney-Stern theorem.Table 1 lists the parameters used for the PV distributions in the different experiments in this study. Because the model describes inviscid, unforced motion, PV is rearranged without any new PV generation in all experiments. Hence, it is of interest to see the redistribution of the same amount of PV under varying basic-state conditions. The total integrated PV over the model domain is to be nearly equal, by design, in all experiments; therefore, the maximum amplitudes of PV rings located at larger radii are smaller. All the model vortices have elevated PV rings of 6 km in width but have different central radii of 18 km, 45 km, and 90 km to mimic real TCs of different eyewall sizes. Although more experiments can be done by further varying the widths, amplitudes, and ring sharpnesses, we will focus on a small parameter space to address the key scientific questions mentioned in section 1. Several three-point smoothing operations are performed to reduce the computational modes related to the sharp PV gradients at both the inner and outer edges of PV rings. As a result, the maximum amplitudes of PV rings in all experiments are reduced slightly, and the uniform PV distributions of eyewall are deconstructed due to the limited spatial resolution of 3 km (Fig. 1). Nevertheless, the general characteristics of the radial PV distributions are conserved, and the constraint of Charney-Stern theorem is still met since the PV structure with respect to the off-centered maximum is kept unchanged. Referring to the segmentation function concerning the basic-state relative vorticity in Schubert et al. (1999),
$ {\xi}_{\text{1}} $ +$ {\text{}\xi}_{\text{2}} $ and$ {\xi}_{\text{2}} $ represent the relative vorticity in the axisymmetric eye (0$ \leqslant $ r$\leqslant{{r}}_{\text{1}}$ ) and eyewall (${{r}}_{\text{1}}\leqslant$ r$\leqslant{{r}}_{\text{2}}$ ) regions, respectively. Two parameters γ = ($ {\xi}_{\text{1}} $ +$ {\xi}_{\text{2}} $ )/$ {\xi}_{\text{ave}} $ ($ {\xi}_{\text{ave}} $ is the average relative vorticity over the region 0$ \leqslant $ r$\leqslant{{r}}_{\text{2}}$ ) and δ =${{r}}_{1}$ /${{r}}_{2}$ are used to measure the contrast of relative vorticity between the central region and the enhanced ring and the degree of hollowness, respectively. In the shallow-water case, we replace relative vorticity with PV. Since the initial PV in the eye region has been set to zero, all the vortices have γ = 0 but differ in terms of δ. Here the expression of δ is given as${{r}}_{\text{i}}$ /${{r}}_{\text{e}}$ , where${{r}}_{\text{i}}$ is the radius of maximum basic-state PV gradient (positive) and${{r}}_{\text{e}}$ is the radius of the minimum basic-state PV gradient (negative). The smoothed PV profile is used to calculate this parameter shown in Table 1.Expt. Maximum PV before
smoothing (${{q} }_{\text{r0} }$, PVU)Maximum PV after
smoothing (PVU)Annulus width
(l, km)ro (km) $\delta $ Coriolis parameter
(in the northern hemisphere)F18 8.21 7.20 6.0 18 0.40 f(15°), $ \beta{= 0} $ F45 3.48 2.86 6.0 45 0.73 f(15°), $ \beta{= 0} $ F90 1.78 1.69 6.0 90 0.88 f(15°), $ \beta{= 0} $ B18 8.21 7.20 6.0 18 0.40 f(15°), $ \beta $(15°) B45 3.48 2.86 6.0 45 0.73 f(15°), $ \beta $(15°) B90 1.78 1.69 6.0 90 0.88 f(15°), $ \beta $(15°) B90-60 1.78 1.69 6.0 90 0.88 f(15°), $ \beta $(60°) B90-75 1.78 1.69 6.0 90 0.88 f(15°), $ \beta $(75°) Table 1. Description of the eight experiments in this study. The second and third columns give information on the maximum potential vorticity (PV) before and after smoothing; the fourth, fifth, and sixth columns present the widths of the PV rings, central radii of the location, and degree of hollowness, respectively. The last column shows the Coriolis parameters used in simulations. The first three experiments are conducted on the f-plane, while the last five are on the beta-plane.
Figure 1. Radial distributions of the initial axisymmetric PV annuli (units: 0.1 PVU) given in Eq. (13) before smoothing (solid) and after smoothing (dashed) for three cases with an ro of 18 km (a), 45 km (b), and 90 km (c), respectively.
It is difficult to determine whether the basic-state flow fulfills the constraint of the Fjortoft theorem because the term on the right-hand side of Eq. (8) includes unknown parameters related to the perturbation. However, the integral of the part that only includes the basic-state parameters, given by
can help determine the sign of the entire integral on the right-hand side of Eq. (8). Negative values (–3400 for 18 km; –1073 for 45 km; –308 for 90 km) for Eq. (14) suggest that all the experiments are likely to satisfy the requirement of the Fjortoft theorem.
Given the basic-state PV distribution, the basic-state wind field is obtained by solving Eq. (12) iteratively, together with the geopotential height obtained from the nonlinear balance equation, which yields the relationship between the geopotential and streamfunction fields as given below
where ψ is the streamfunction. Equation (15) is numerically solved using a simple over-relaxation method based on the basic-state wind field (streamfunction) to obtain the initial geopotential height in Eq. (12). Ten iterations are deemed sufficient to obtain convergent numerical solutions.
Three f-plane experiments are performed to serve as a basis for comparison with results from other experiments that include the beta effect (Table 1). Note that we initialize the model with a vortex structure that can enable the barotropic instability, which may develop freely with even small model numerical truncation errors. The opposite signed sharp radial PV gradients at radii of ro ± l/2 permit the emergence of a pair of azimuthally counter-propagating Rossby edge-waves (Schubert et al., 1999). In contrast, the uniform distribution of PV within the interior and beyond the exterior boundary of the annulus is unfavorable for wave generation in our simulations.
Expt. | Maximum PV before smoothing (${{q} }_{\text{r0} }$, PVU) | Maximum PV after smoothing (PVU) | Annulus width (l, km) | ro (km) | $\delta $ | Coriolis parameter (in the northern hemisphere) |
F18 | 8.21 | 7.20 | 6.0 | 18 | 0.40 | f(15°), $ \beta{= 0} $ |
F45 | 3.48 | 2.86 | 6.0 | 45 | 0.73 | f(15°), $ \beta{= 0} $ |
F90 | 1.78 | 1.69 | 6.0 | 90 | 0.88 | f(15°), $ \beta{= 0} $ |
B18 | 8.21 | 7.20 | 6.0 | 18 | 0.40 | f(15°), $ \beta $(15°) |
B45 | 3.48 | 2.86 | 6.0 | 45 | 0.73 | f(15°), $ \beta $(15°) |
B90 | 1.78 | 1.69 | 6.0 | 90 | 0.88 | f(15°), $ \beta $(15°) |
B90-60 | 1.78 | 1.69 | 6.0 | 90 | 0.88 | f(15°), $ \beta $(60°) |
B90-75 | 1.78 | 1.69 | 6.0 | 90 | 0.88 | f(15°), $ \beta $(75°) |