In this study, the effect of the shape of basic-state potential vorticity (PV) profile on the stability of the tropical cyclone (TC)-like vortices possessing hollow PV structure is discussed in a two-dimensional (2D) shallow water barotropic model. Under the premise of the same basic-state maximum wind and the radius of maximum wind (RMW), a sequence of 170 numerical simulations are conducted by setting two structural parameters of basic-state PV, with the first parameter defining the thickness of the ring (i.e., the ratio of the inner and outer radii of the ring) and the second parameter defining the hollowness of the ring (i.e., the ratio of eye to inner-core relative vorticity). The eigenfrequency and growth rate of the most unstable mode in systems (MUMSs) as well as the most unstable wavenumber (MUWN) with different the basic-state PV profiles are examined. The results show that (1) the MUWN for thicker rings are more prone to lower wavenumber (WNs), and the corresponding MUMSs possess lower intrinsic frequencies and small growth rates; (2) thinner rings are more prone to higher wavenumber growth, and the more filled the rings become, the higher wavenumber (WN) the MUWN; and (3) for thin and hollow rings, MUMSs possess high frequencies and large growth rates. However, in real cases, the structure and intensity of the basic state vortex are changing at any time during the development of TCs. Thus, in order to make the study more complete, sensitivity of intensity of the basic state vortex to the stability of system is further conducted by setting three different hollow PV structures in a two-dimensional (2D) shallow water barotropic model. The results show that the intensity of the basic state vortex mainly affects the strength of the instability, that is the disturbance growth rates become larger with an increase of the strength of vortices in the same PV structure, while the shape of the PV profile is the key factor to determine the stability of a vortex.
Considering that the instability is selective for the WNs, the instability properties and disturbances development of different azimuthal wavenumbers (WNs) are analyzed. For the thick and filled PV rings, the weak instability is more likely to occur at lower WNs, and the wavenumber one disturbance appears to be algebraically growing. For the moderate thick and hollow PV rings, the perturbation kinetic energy at the MUWN increases exponentially and the disturbance growth of the MUWN is much faster than that of other WNs. For the thin and hollow PV rings, the disturbance development depends mainly on the instability of higher WNs. And the growth rates of the fastest growing mode (largest dimensionless growth rate) for two or more different WNs are very close, which causes the disturbance growth of multiple WNs are approaching. The dynamics of the asymmetric structure in the inner core of TC-like vortices are further advanced. It is found that for the weakly unstable rings, the wave structure of MUMSs have typical characteristics of Vortex Rossby Waves (VRWs), while for the strongly unstable rings, the wave structure of MUMSs possess typical properties of the mix waves.
To confirm the above results, a high resolution dataset from a realistic simulation is used to examine the impacts of the evolution of PV ring on intensity and structure change of TCs. It is shown that the polygonal features appearing in the TC eyewall are highly consistent with the MUWN determined by the radial distribution of the basic state PV at the current time. This indicates that the basic state PV profile is selective to the most unstable growth of WN, which is beneficial to the disturbance growth at the MUWN. Moreover, the dataset also demonstrates the dynamic instability with the evolution of PV ring is closely related with the development stage of TC.