When is a sloshing vortex an analogue black hole bomb?

TL;DR

The paper addresses the stability of a shallow-water Rankine vortex with a free surface, using a Clebsch-based variational framework that accommodates both irrotational and rotational perturbations. By deriving a conserved energy and introducing gauge-invariant perturbations, it isolates negative-energy modes localized in the vortex core and studies their coupling to external surface waves. Through analytical characterization of WP, DPR, and DPP regimes and numerical methods (shooting and Leaver’s continued fractions) for $m=2$ and higher, it reveals a transition from vorticity-driven instabilities at low circulation to ergoregion (black-hole–like) instabilities at high circulation, with hybrid instabilities arising in finite containers. The results identify hollow-core or dry-patch vortices as optimal regimes for exploring analogue-gravity instabilities and have implications for laboratory tests of superradiance and for understanding related astrophysical black hole bomb phenomena.

Abstract

Draining vortices provide a powerful platform for simulating black hole phenomena in tabletop experiments. In realistic fluid systems confined within a finite container, low-frequency waves amplified by the vortex are reflected at the walls, rendering the system unstable. This process, known in the gravitational context as the black hole bomb, manifests as a sloshing motion of the free surface. The analogy, however, becomes more nuanced when a realistic vortex core with a non-singular vorticity distribution is considered. We investigate this by analysing a non-draining Rankine vortex in the shallow-water and inviscid limits. At low circulation, the sloshing corresponds to an instability of the vorticity field, whereas at high circulation where fluid is expelled from the vortex core, the destabilising mechanism coincides with that of the black hole bomb. Our variational framework distinguishes the energetic contributions of vorticity and irrotational perturbations, offering new insight into the rotating-polygons instability reported by, e.g. Jansson et al. (2006). From the analogue-gravity perspective, we identify hollow core vortices as an optimal regime for exploring black-hole-like instabilities in fluids.

Figures (9)

• Figure 1: Various types of superradiant instabilities in irrotational swirling flows. Top: the black hole bomb (BHB) occurs as a positive-energy state which is trapped outside the vortex, between the effective potential barrier representing the analogue ergoregion (red; see § \ref{['ssec:lagr-perturbations']}) and the outer boundary. It grows because negative energy is transmitted into the interior and dissipated, e.g. absorbed at the event horizon. Middle: the ergoregion instability is a negative-energy state in the interior region, which grows when positive energy is radiated to infinity. Bottom: if the system is closed on both ends, a hybrid instability occurs when the inner and outer states resonate with each other. Although the three types of instability differ by which region of space they are localised in, they all rely on energy transfer across the ergoregion.
• Figure 2: Cross section of the non-draining vortex in the $(x,z)$ plane to which we apply our inviscid model. The domain is axially symmetric and, in some cases, spatially constrained by a solid boundary at $r = L$.
• Figure 3: Background flow field solutions for four different $C$ values in non-dimensional units, see Eq. \ref{['adim']}. The black (red) curve is $h$ ($v_\theta^2$) as a function of $r$. The dashed black line is the boundary of the rotational core of the vortex. The pink shaded region represents the ergoregion where $v_\theta^2 > h$ (the flow exceeds the wave speed), whilst the grey shaded region represents the dry patch. (a): at low $C$, there is no ergoregion and the rotational core covers the whole central region. (b): as $C$ increases, an ergoregion develops around the edge of the rotational core. (c): increasing $C$ further, a dry patch develops in the centre. (d): for high enough $C$, the boundary of the dry patch is in the potential region ($r > 1$) and the rotational core is absent from the flow.
• Figure 4: Eigenfrequencies of the unstable $m=2$ mode (solid black lines) as a function of $C$ for the open system. The oscillation rate $\mathrm{Re}[\omega]$ is shown in panel (a), while the growth rate $\mathrm{Im}[\omega]$ is presented in panel (b). The dashed vertical lines indicate the boundaries of different flow regimes. At low $C$, the flow is wet-plate (WP). An ergoregion is present everywhere inside the pink region. As we increase $C$, the flow becomes dry-plate rotational (DPR) at the first black dashed line and dry-plate potential (DPP) at the second one. The dashed red lines indicate the $1/C$ fall off at large $C$ and the linear dependence at low $C$ in the real part.
• Figure 5: Energy components of the $E<0$ mode with $m=2$ for $L=3.5$. In the WP regime, $E_\mathrm{rot}$ is the dominant energy component and the eigenmode is due to an excitation of the vorticity field. In the DPP regime, the flow is potential and the negative energy of this state is the result of the ergoregion. The presence of an ergoregion is indicated by the shaded pink area. The inset (b) shows the mode frequency, which resembles $\mathrm{Re}[\omega]$ in the case of the open system [see Fig. \ref{['fig:4']}(a)], with the deviations from the expected $1/C$ behaviour at large $C$ resulting from finite size effects.