Black hole spectral instabilities in the laboratory: Shallow water analogue

Small deviations in the spacetime around black holes can lead to instabilities in the underlying quasinormal mode spectrum, potentially altering the hierarchy of its overtones. A practical way to induce such spectral instability is by introducing small modifications to the effective potential governing the dynamics of fluctuations in the black hole spacetime. While finding a physically meaningful interpretation for such ad hoc modifications in an astrophysical context can be challenging, analogue black hole models provide an alternative framework to explore their effects and study the instabilities. In this work, we consider an analogue black hole modeled by a draining bathtub flow and demonstrate that vorticities in the fluid introduce a small bump in the effective potential of the wave equation. This naturally realizes a physically motivated version of the elephant and the flea configuration. We analyze the spectrum using two complementary approaches: direct mode computation via two distinct frequency-domain methods and time evolution of initial perturbations. As in astrophysical black holes, the vorticities destabilizes the QNM spectrum of the analogue system, possibly yielding time evolution with long-lived ringing effects, akin to those observed for massive fields in curved spacetimes.