## Results

We first diagnose the result from the wavenumber 2 perturbation simulations. Figure 4 shows the time evolution of the simulated symmetric tangential wind profile at a 3-h interval. As one can clearly see, a double-peak wind profile appears at hour 6. After that, the outer maximum continues to grow, while the inner peak experiences an oscillation in amplitude. For instance, at hour 9, there are two peaks in the symmetric tangential wind profile, with the inner one retaining bigger amplitude; at hour 12, the outer one has stronger amplitude.

A key question related to this double eyewall formation is how the outer wind peak is established. A notable feature is that the symmetric tangential wind in the outer region (r > 0.15) continues to grow while the wind in the inner region oscillates after initial rapid decay. These distinctive evolution features between the outer and inner regions are closely related to the energy transfer between the symmetric flow and the asymmetric perturbation, as shown in Fig. 5.

The diagnosis of the energy exchange between the symmetric and asymmetric components (the second-to-fifth terms on right-hand side of (2.4)) shows that outside of r = 0.15 there is always a positive energy

OOhr 03hr

Radius

Fig. 4. The evolution of non-dimensional symmetric tangential wind profiles for case T20. To obtain tangential wind in ms_1, multiply by 50. To obtain radial displacement in km, multiply by 1000.

Sym. KE rate by wave-wave

Sym. KE rate by wave-wave

Radius

Fig. 5. The time—radius cross-section of the asymmetry-to-symmetric kinetic energy transfer rate (unit: 1.25 X 10-4 m2s-3) in association with the wave—wave interactions in case T20. To obtain radial displacement in km, multiply by 1000. The time unit is hour.

Radius

Fig. 5. The time—radius cross-section of the asymmetry-to-symmetric kinetic energy transfer rate (unit: 1.25 X 10-4 m2s-3) in association with the wave—wave interactions in case T20. To obtain radial displacement in km, multiply by 1000. The time unit is hour.

transfer from the asymmetric perturbation to the symmetric flow, whereas inside of this radius there is oscillatory behavior in the energy transfer, that is, the symmetric flow gains energy from the asymmetry during hours 4-9 but loses energy into the asymmetry during hours 0-4 and 9-12. This is consistent with the time tendency of the symmetric tangential wind near RMW.

To understand the cause of the distinctive energy transfer behavior, we examine the asymmetric perturbation structure and its evolution characteristics. Figure 6 shows the time evolution of amplitude of the asymmetric perturbation. Note that in this numerical experiment (T20) the initial wavenumber 2 asymmetry is placed at the radius of 0.2. After time integration, a strong asymmetry is generated within the first four hours inside the radius of 0.1 where the absolute value of the symmetric vorticity gradient is the largest (Fig. 2(d)). The asymmetry amplitude in the outer region (r > 0.15), however, decreases gradually.

The horizontal pattern of the asymmetric perturbation reveals that the asymmetric vorticity field exhibits distinctive patterns during the different development stages. For example, at hour 1, the phase line connecting this newly generated asymmetry inside of RMW and the original outer

Asymétrie vorticity h

Asymétrie vorticity h

Radius

Fig. 6. The time—radius cross-section of the asymmetric vorticity amplitude (unit: 5 X 10_5 s~for case T20. To obtain radial displacement in km, multiply by 1000. The time unit is hour.

Radius

Fig. 6. The time—radius cross-section of the asymmetric vorticity amplitude (unit: 5 X 10_5 s~for case T20. To obtain radial displacement in km, multiply by 1000. The time unit is hour.

asymmetry shows an up-shear tilt (Fig. 7(a)) with respect to the rotation angular velocity of the core vortex (Fig. 2(b)). Because of this up-shear tilt, the symmetric flows transfer their energy to the asymmetric perturbations near r = 0.1 before hour 4 (Fig. 5), resulting in the weakening of the symmetric core vortex at hour 3 (Fig. 4). Because the symmetric angular velocity advects the inner asymmetry at a much faster rotation rate (see the angular velocity profile in Fig. 2(b)) than the outer asymmetry, the asymmetric vorticity shifts its phase to a down-shear tilt in the period of hours 4-9 (refer to Fig. 7(b)), so that the energy is transferred back to the symmetric flows (Fig. 5) and the asymmetric vorticity amplitude decreases during the period (Fig. 6). Thus, the symmetric tangential wind near the radius of 0.1 grows at the expense of the weakening of the asymmetry from hour 6 to hour 9 (Fig. 4). A new up-shear-tilting inner asymmetry is induced again after hour 9 (Figs. 6 and 7(c)). As a result, the symmetric flows transfer their energy to the asymmetric perturbations during hours 9-12, while the tangential wind at the inner core region weakens (Fig. 4).

In contrast, the symmetric flows always gain energy from the asymmetric disturbances in the outer region (r > 0.15) due to the steady down-shear phase tilt of the asymmetric disturbances (Fig. 7). This causes

Asymmetric Vorticity Time=01hr

Asymmetric Vorticity Time=05hr

Asymmetric Vorticity Time=01hr

Asymmetric Vorticity Time=05hr

180 270

### Tangential

Fig. 7. The non-dimensional asymmetric vorticity pattern at time (a) 01 h, (b) 05 h and (c) 12h for case T20. To obtain vorticity in s~1, multiply by 5 X 10~5. To obtain radial displacement in km, multiply by 1000.

180 270

### Tangential

Fig. 7. The non-dimensional asymmetric vorticity pattern at time (a) 01 h, (b) 05 h and (c) 12h for case T20. To obtain vorticity in s~1, multiply by 5 X 10~5. To obtain radial displacement in km, multiply by 1000.

the continuous intensification of the tangential wind in the outer region, leading to the formation of the second peak in the symmetric tangential wind profile.

To examine whether the aforementioned wind evolution characteristics change with different initial perturbations, we conduct a set of parallel experiments in which an initial wavenumber 3 asymmetry is introduced. Figure 8 shows the symmetric tangential wind evolution in case H20, where the initial asymmetric perturbation is placed at the radius of 0.2. Compared to case T20, a weaker asymmetry is generated near the radius of 0.1 (Fig. 9(a)). The comparison of symmetric kinetic energy change rates between the wavenumber 2 (T20) and wavenumber 3 (H20) perturbation

Radius

Fig. 8. The evolution of non-dimensional symmetric tangential wind profiles for wavenumber 3 initial disturbances in case H20. To obtain tangential wind in ms_1, multiply by 50. To obtain radial displacement in km, multiply by 1000.

### Radius

Fig. 8. The evolution of non-dimensional symmetric tangential wind profiles for wavenumber 3 initial disturbances in case H20. To obtain tangential wind in ms_1, multiply by 50. To obtain radial displacement in km, multiply by 1000.

Fig. 9. The time—radius cross-section of (a) the asymmetric vorticity amplitude (unit: 5 X 10_5 s~x) and (b) the asymmetry-to-symmetry kinetic energy transfer rate (unit: 1.25 X 10_4 m2 s~3) for wavenumber 3 initial disturbances in case H20. To obtain radial displacement in km, multiply by 1000. The time unit is hour.

experiments indicate that the energy exchange between the asymmetric and symmetric flows is weaker in the wavenumber 3 case. Nevertheless, a weak oscillation of the energy transfer is still present near the radius of 0.1, while in the outer region (r > 0.15) the wavenumber 3 initial disturbance can transfer more energy into the symmetric flows than the corresponding

Fig. 10. The non-dimensional symmetric tangential wind profiles at 12 h for (a) wavenumber 2 and (b) wavenumber 3 initial disturbances. To obtain tangential wind in ms_1, multiply by 50. To obtain radial displacement in km, multiply by 1000.

wavenumber 2 disturbance (Fig. 9(b)). It is again the down-shear tilt of the asymmetric perturbation and so induced asymmetry-to-symmetry energy transfer in the outer region that generate the second peak in the symmetric tangential wind profile (Fig. 8).

For the same-structure initial perturbation, is there a preferred radius location for the double eyewall formation? Our sensitivity experiments with the same wavenumber 2 or 3 initial perturbation but with different radial locations show that indeed there exists such an optimal radius. When the perturbation is placed more outward (i.e. T25, T30, H25, and H30) compared to the T20 and H20 experiments, the second peak in the symmetric tangential wind profile becomes weaker (Fig. 10), which means that the symmetric flows gain less energy from the asymmetric perturbations. On the other hand, when the initial asymmetry is placed more inward in T15 and T10 (H15 and H10), there is no obvious second-peak in the tangential wind profile. Thus, the sensitivity experiments above point out an optimal location near r = 0.2 (i.e. twice of RMW), where the initial asymmetry may generate the most significant double peaks in the symmetric tangential wind profile (Fig. 10).

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