Amplitude |𝒬| (solid lines) and normalised phase φ_𝒬/2π (dashed lines) of the complex amplification factor associated through Eqs. (86), (87) to the fundamental eigenfrequency ω calculated with cooling (purple lines with crosses) and in the adiabatic approximation (green lines), using the same adiabatic estimate of the advection time ${\tau }_{\mathrm{adv}}^{\mathrm{ns}}$ as that in Eq. (79). The orange line shows |𝒬| and φ_𝒬/2π deduced from the analytic estimate (orange lines) using Eq. (82) with ω satisfying Eq. (83).