  Analogue systems are a powerful instrument to investigate and understand in a
controlled setting many general-relativistic effects. Here, we focus on
superradiant-triggered instabilities and quasi-normal modes. We consider a
compressible hydrodynamic vortex characterized by a polytropic equation of
state, the {\it polytropic hydrodynamic vortex}, a purely circulating system
with an ergoregion but no event horizon. We compute the quasinormal modes of
this system numerically with different methods, finding excellent agreement
between them. When the fluid velocity is larger than the speed of sound, an
ergoregion appears in the effective spacetime, triggering an "ergoregion
instability." We study the details of the instability for the polytropic
vortex, and in particular find analytic expressions for the marginally-stable
configuration.
