标题： 多态流体动力学涡旋的边缘稳定共振模式 显示英文标题

标题： Marginally stable resonant modes of the polytropic hydrodynamic vortex

摘要： 多级流体动力学涡旋描述了一个有效的$(2+1)$维声学时空，在$r=r_{\text{c}}$处有一个内部反射边界。 这个物理系统，像旋转的Kerr黑洞一样，具有半径为$r_{\text{e}}$的能层和半径为$r_{\text{s}}$的非点状曲率奇点。 有趣的是，表征有效几何的的基本比值$r_{\text{e}}/r_{\text{s}}$仅由循环流体的无量纲多级指数$N_{\text{p}}$决定。 最近证明，在$N_{\text{p}}=0$的情况下，有效声学时空由一个{\it 无限}可数的反射表面半径集合，$\{r_{\text{c}}(N_{\text{p}};n)\}^{n=\infty}_{n=1}$，可以支持静态（边缘稳定）声模。 在本文中，我们使用{\it 分析的}技术来探讨多态流体动力学涡旋在$N_{\text{p}}>0$范围内的物理特性。 特别地，我们证明在这个物理范围内，有效的声学时空由一组{\it 有限}的反射表面半径$\{r_{\text{c}}(N_{\text{p}},m;n)\}^{n=N_{\text{max}}}_{n=1}$表征，这些半径可以支持边缘稳定的静态声模式（此处$m$是声学扰动场的方位调和指数）。有趣的是，已经通过分析证明，无量纲最外层支撑半径$r^{\text{max}}_{\text{c}}/r_{\text{e}}$，它标志着多态流体涡旋中超辐射不稳定性的发生，随着整数调和指数$m$的增加和无量纲多态指数$N_{\text{p}}$的减少而单调增加。 显示英文摘要

摘要： The polytropic hydrodynamic vortex describes an effective $(2+1)$-dimensional acoustic spacetime with an inner reflecting boundary at $r=r_{\text{c}}$. This physical system, like the spinning Kerr black hole, possesses an ergoregion of radius $r_{\text{e}}$ and an inner non-pointlike curvature singularity of radius $r_{\text{s}}$. Interestingly, the fundamental ratio $r_{\text{e}}/r_{\text{s}}$ which characterizes the effective geometry is determined solely by the dimensionless polytropic index $N_{\text{p}}$ of the circulating fluid. It has recently been proved that, in the $N_{\text{p}}=0$ case, the effective acoustic spacetime is characterized by an {\it infinite} countable set of reflecting surface radii, $\{r_{\text{c}}(N_{\text{p}};n)\}^{n=\infty}_{n=1}$, that can support static (marginally-stable) sound modes. In the present paper we use {\it analytical} techniques in order to explore the physical properties of the polytropic hydrodynamic vortex in the $N_{\text{p}}>0$ regime. In particular, we prove that in this physical regime, the effective acoustic spacetime is characterized by a {\it finite} discrete set of reflecting surface radii, $\{r_{\text{c}}(N_{\text{p}},m;n)\}^{n=N_{\text{max}}}_{n=1}$, 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 $r^{\text{max}}_{\text{c}}/r_{\text{e}}$, 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 $N_{\text{p}}$.