Frequency ω_r as a function of the vertical wavenumber k_z of the neutral modes for the four upper branches (panel a) and the four lower branches (panel b) at Ω₀/f = 3 for m = 0 (black), m = 1 (blue), and m = 2 (red). Frequency ω_r (panel c) and growth rate ω_i (panel d), respectively, versus vertical wavenumber k_z for the most unstable mode at Ω₀/f = 3 for m = 0 (black), m = 1 (blue), and m = 2 (red). For all panels, solid lines are numerical results and dashed lines denote the WKBJ predictions for panel a: Eq. (44), panel b: Eq. (45) for m = 0, 1 (core mode) and (A.22) for m = 2 (ring mode), and panels c and d: Eq. (46).