Based on Eq. (17) for wave rays, the radial and tangential group velocities (Cg,r and Cg,θ) are two important factors affecting wave rays. In the following, we take μ=0.2 and θ0=0 as an example to explore the impacts of asymmetric basic flow on the vortex Rossby wave rays via analyzing characteristics of the radial and tangential group velocities.
Figure 4 displays the distributions of dimensionless radial group velocity Cg,r on the θ-r plane in both symmetric and asymmetric basic flows. In the symmetric basic flow (Fig. 4a), Cg,r is always irrelevant to azimuth angle θ and only changes with radius r. The large value of Cg,r appears at r=0.1 (where the maximum tangential wind speed appears). In the asymmetric basic flow (Fig. 4b), Cg,r changes with both r and θ and the largest value also appears at around r=0.1. The average dimensionless value of Cg,r is about 0.50 and the largest is 1.21 within the ranges of \(r\sim[0,0.35]\) and \(\theta\sim[0,2\pi]\), corresponding to an average and maximum radial group velocity of 25 m s-1 and 60.5 m s-1. With Eq. (8), it can be found that when μ=0.2 the dimensionless maximum radial basic flow velocity \(\bar{u}\) is -0.16, corresponding to -8.0 m s-1, which is much slower than the maximum Cg,r. The average dimensionless radial basic flow speed is -0.14, corresponding to the average value of -6.8 m s-1, which is also much slower than the average Cg,r. When compared with \(\bar{v}\), although the average Cg,r is smaller, the maximum value of Cg,r is almost the same as \(\bar{v}\). This indicates that the characteristic of Cg,r is identical to that of \(\bar{v}\), and the wave energy could propagate outward rapidly.
The difference between radial group velocities in symmetric and asymmetric basic flows can be written as \begin{equation} \label{eq17} \Delta C_{g,r}=C_{g,r_{\rm asy}}-C_{g,r{\rm _sy}} , \ \ (19)\end{equation} where Cg,r _asy and Cg,r _sy are radial group velocities in symmetric and asymmetric conditions, respectively.
The distribution of ∆ Cg,r on the θ-r plane is displayed in Fig. 5, which shows that ∆ Cg,r also reaches its maximum at around r=0.1.
It is worth noting that the distributions of Cg,r and ∆ Cg,r shown in Fig. 4 and Fig. 5 are obtained under the condition of an initial azimuth angle θ0=0 in the tangential wavenumber-1 perturbation flow and the radial basic flow. Compared with Fig. 1, it is easy to identify that ∆ Cg,r is affected by both the features of asymmetry in the tangential and radial basic flows. For example, the maximum and minimum values of the wavenumber-1 perturbation flow are located at azimuth angles of θ =0.5π and 1.5π, and the locations of maximum and minimum values of the radial basic flow are at a θ=π and 0. When r is small, the maximum and minimum values of ∆ Cg,r appear at θ≈π and 2π, which is close to the location of the maximum and minimum values of \(\bar{u}\). Whereas, for the larger r, the location of the maximum and minimum values of ∆ Cg,r will rotate clockwise and approach those of \(\bar{v}\), as shown in Fig. 5. This suggests that both the tangential asymmetric wavenumber-1 flow and the radial basic flow can affect the distribution of radial group velocity Cg,r; however, the tangential and radial basic flow shows different importance for different r. Generally, the locations of the maximum and minimum value of ∆ Cg,r are closely linked with the azimuth angles of the asymmetry. Therefore, if the asymmetries of the basic flows intensify at specific azimuth angles, then Cg,r will increase around these angles, and the wave energy will propagate outward more rapidly there. Similarly, when the asymmetry decreases at specific azimuth angles, the wave energy will propagate outward more slowly along these directions.
The distribution of tangential group velocity Cg,θ on the θ-r plane is presented in Fig. 6, which shows that the distribution of Cg,θ (Fig. 6a) in the symmetric basic flow is similar to that of Cg,r (Figure 4a), and both change only with radius and reach their maximum values at r=0.1. In the asymmetric basic flow, however, although the maximum values of Cg,r and Cg,θ both appear at around r=0.1 (Fig. 6b and Fig. 4b), with little difference in values, Cg,θ is much larger than Cg,r at a certain radius (e.g., at r=0.3, Cg,r≈ 0.2, Cg,θ≈ 0.6), indicating that Cg,r and Cg,θ are different from each other. Cg,r decreases more rapidly along the radial direction, and thereby the speed of wave energy propagation along the radial direction will become slow beyond a certain radius. In contrast, the radial gradient of Cg,θ is relatively small, so the tangential propagation of wave energy can still maintain in regions far away from the typhoon center. It can be estimated that the average value of dimensionless Cg,θ in the region shown in Fig. 6 \((r\sim [0,0.35],\theta\sim[0,2\pi]\)) is 0.75, which corresponds to 37.5 m s-1, and is just the same as the average value of \(\bar{v}\). Whereas, the maximum value of dimensionless Cg,θ is 1.20, corresponding to 60.0 m s-1, which is also the same as the maximum value of \(\bar{v}\). This suggests that wave energy can propagate along the tangential direction at the average speed of the tangential basic flow. In short, the wave energy propagation along the tangential direction is much faster than that in the radial direction.
The difference between tangential group velocities in symmetric and asymmetric basic flows (∆ Cg,θ) can be expressed as \begin{equation} \label{eq18} \Delta C_{g,\theta}=C_{g,\theta_{\rm asy}}-C_{g,\theta_{\rm sy}} ,\ \ (20) \end{equation} where Cg,θ_ asy and Cg,θ_ sy are the tangential group velocities in symmetric and asymmetric basic flows, respectively.
Figure 7 shows the distribution of ∆ Cg,θ on the θ-r plane. ∆ Cg,θ changes with both azimuth angle θ and radius r. The change in ∆ Cg,θ with θ demonstrates the asymmetric feature of tangential wavenumber-1 flow and the radial basic flow, and the azimuth angles corresponding to the locations of asymmetric maximum and minimum values also change with r.
Compared with Fig. 1, it is clear that when r is relatively small, the large and small values of ∆ Cg,θ are located just between the locations of the maximum and minimum value of \(\bar{v}\) and \(\bar{u}\) (e.g., when \(r\sim[0.01,0.05]\), the maximum and minimum values of ∆ Cg,θ are located at about 0.3π and 1.3π), which means that the basic flow \(\bar{v}\) and \(\bar{u}\) are both important at this radial scope. However, when r becomes larger, areas of large and small values of ∆ Cg,θ turn anticlockwise and basically overlap areas of large and small values of wavenumber-1 perturbation flow (e.g., when r≈ 0.1, the large and small ∆ Cg,θ are located at about 0.5π and 1.5π), which suggests that the tangential wavenumber-1 perturbation plays a more important role for such conditions. Moreover, the locations of large and small ∆ Cg,θ turn anticlockwise with a continuous increase of r, indicating that the influence of radial flow decreases rapidly with increasing r. As a result, both the tangential wavenumber-1 perturbation flow and the radial basic flow both can increase (decrease) the value of ∆ Cg,θ, but the importance of the radial basic flow is only concentrated in areas for smaller r, and ∆ Cg,θ is dominated by the tangential wavenumber-1 perturbation when r becomes larger.
Based on the wave ray equation, Eq. (17), the ratio of Cg,r to Cg,θ determines the ray slope of vortex Rossby waves. Distributions of ray slopes in symmetric and asymmetric basic flows are shown in Fig. 8, which indicates that in the symmetric basic flow (Fig. 8a) the ray slope is only associated with the radius, and its distribution is similar to that of the group velocity, with the maximum value occurring at r=0.1. In the asymmetric basic flow (Fig. 8b), the distribution of the ray slope is completely different to that of the group velocity, although the large ray slope still occurs at around r=0.1. Impacts of asymmetry in the basic flow on the wave ray slope are mainly concentrated near the RMW, and the azimuth angle corresponding to the maximum wave ray slope is at around θ=1.13π. Differences between the ray slopes in symmetric and asymmetric basic flows (Fig. 9) indicate that large differences also occur around the RMW, whereas the azimuth angle corresponding to the largest difference varies with the maximum speed of tangential wavenumber-1 perturbation flow and the radial basic flow. It is worth noting that the influence of asymmetry is still important for larger r, and this can explain the reason why the difference between wave rays of asymmetric flow and symmetric flow is great in Fig. 3.