A nondivergent barotropic model (Model 1) and a barotropic primitive equation vortex model (Model 2) are linearized respectively in this paper. Then their perturbation wave spectrums are computed with a normal mode approach to study the instability problem on an appointed tropical cyclone (TC)-like vortex, thereby, the dynamic instability properties of spiral cloud bands of TCs are discussed. The results show that the unstable mode of both models exhibits a spiral band-like structure that propagates away from the vortex outside the radius of maximum winds. The discrete modal instability of the pure vortex Rossby wave can account for the generation of the eyewall and the inner spiral band. The unstable mode in Model 2 has three parts, i.e., eyewall, inner and outer spiral bands. This mode can be interpreted as a mixed vortex Rossby-inertia gravitational wave. The unbalanced property of the wave outside the stagnation radius of the vortex Rossby wave is one of the important reasons for the formation of the outer spiral band in TCs. Accordingly, the outer spiral band can be identified to possess properties of an inertial-gravitational wave. When the formation of unstable inner and outer spiral bands is studied, a barotropic vortex model shall be used. In this model, the most unstable perturbation bears the attributes of either the vortex Rossby wave or the inertial-gravitational wave, depending on the vortex radius. So such perturbations shall be viewed as an unbalanced and unstable mixed wave of these two kinds of waves.
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