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In this study, the linearized, f-plane, shallow-water equations are discretized into a matrix eigenvalue problem to examine the full spectrum of free waves on barotropic (monopolar and hollow) vortices. A typical wave spectrum for weak vortices shows a continuous range between zero and an advective frequency associatedwith vortexRossby waves (VRWs)and two discreteranges at both sides associatedwith inertio-gravity waves (IGWs). However, when the vortex intensity reaches a critical value, higher-frequency waves will be ‘‘red shifted’’ into the continuous spectrum, while low-frequency waves will be ‘‘violet shifted’’ into the discrete spectrum, leading to the emergence of mixed vortex Rossby‐inertio-gravity waves (VRIGWs). Results show significant (little) radial wavelike structures of perturbation variables for IGWs (VRWs) with greater (much smaller) divergence than vorticity and the hybrid IGW‐VRW radial structures with equal amplitudes of vorticity and divergence for mixed VRIGWs. In addition, VRWs only occur within a critical radiusat whichtheperturbationazimuthalvelocityisdiscontinuous. As theazimuthalwavenumber increases, lower-frequencywavestendtoexhibitmoremixed-wavecharacteristics,whereashigher-frequencywaveswill be more of the IGW type. Two-dimensional wave solutions show rapid outward energy dispersion of IGWs and slower dispersion of VRWs and mixed VRIGWs in the core region. These solutions are shown to resemble the previous analytical solutions, except for certain structural differences caused by the critical radius. It is concluded that mixed VRIGWs should be common in the eyewall and spiral rainbands of intense tropical cyclones. Some different wave behaviors associated with the monopolar and hollow vortices are also discussed.

An Eigenfrequency Analysis of Mixed Rossby–Gravity Waves on Barotropic Vortices