Marginally stable resonant modes of the polytropic hydrodynamic vortex

Abstract

The polytropic hydrodynamic vortex describes an effective (2+1)-dimensional acoustic spacetime with an inner reflecting boundary at r=rc. This physical system, like the spinning Kerr black hole, possesses an ergoregion of radius re and an inner non-pointlike curvature singularity of radius rs. Interestingly, the fundamental ratio re/rs which characterizes the effective geometry is determined solely by the dimensionless polytropic index Np of the circulating fluid. It has recently been proved that, in the Np=0 case, the effective acoustic spacetime is characterized by an infinite countable set of reflecting surface radii, {rc(Np;n)}n=1n=∞, that can support static (marginally-stable) sound modes. In the present paper we use analytical techniques in order to explore the physical properties of the polytropic hydrodynamic vortex in the Np>0 regime. In particular, we prove that in this physical regime, the effective acoustic spacetime is characterized by a finite discrete set of reflecting surface radii, {rc(Np,m;n)}n=1n=Nmax, that can support the marginally-stable static sound modes (here m is the azimuthal harmonic index of the acoustic perturbation field). Interestingly, it is proved analytically that the dimensionless outermost supporting radius rcmax/re, which marks the onset of superradiant instabilities in the polytropic hydrodynamic vortex, increases monotonically with increasing values of the integer harmonic index m and decreasing values of the dimensionless polytropic index Np.