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Quasinormal modes of the polytropic hydrodynamic vortex

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.


I. INTRODUCTION
Black holes are an important component of our Universe, thought to play a role in the dynamics of galaxies and in star formation. They have also come to play a prominent role in high-energy physics, and even fundamental physics as they are simultaneously an "elementary particle" of gravitation and an entity where classical and quantum effects are intertwined through Hawking radiation. Unfortunately, many of these features and properties have the undesirable consequence that black holes are "hard to see". It is therefore very convenient to use analogue setups where these properties are present, but which can be manipulated in the laboratory. One such example are acoustic holes, idealized models in fluid dynamics mimicking the behavior of a curved spacetime [1][2][3].
Acoustic holes have, for instance, been used to understand wave scattering in more general curved spacetimes [4][5][6][7][8] as well as the impact of UV cutoffs in the Hawking radiation in reasonable completions of General Relativity [9]. One specially interesting feature of curved spacetimes in General Relativity is the ergoregion, bounded by an infinite-redshift surface which represents the static limit where it's impossible to remain at rest with respect to distant inertial observers. Negativeenergy states are possible within the ergoregion, leading to superradiant effects if horizons are present and to instabilities otherwise [10].
The purpose of this work is to explore the phenomenology of superradiant instabilities in analogue models. We focus on the hydrodynamic vortex [11][12][13], a purely circulating compressible system, and we will work with a polytropic equation of state [14].
Here we show that the polytropic hydrodynamic vortex is unstable under linearized perturbations, and that the appearance of these instabilities is directly related to the presence of an ergoregion and absence of an event horizon, the so-called ergoregion instability [10,15]. This result is an generalization of a previous result (obtained for an incompressible system in [16]) for compressible systems that satisfy a polytropic equation of state and that describe a wide-class of thermodynamical processes, as, e.g., isentropic processes [17][18][19][20]. Here we focus on values of polytropic index N p describing isentropic processes (also adiabatic processes), with a compatible experimental setup in perfect gases.
The remainder of this paper is structured as follows. In Sec. II we describe the spacetime of the polytropic hydrodynamic vortex. In Sec. III we study the perturbations of the polytropic hydrodynamic vortex using descriptions in the time and frequency domains. In Sec. IV we obtain the quasinormal mode (QNM) frequencies of the polytropic hydrodynamic vortex using the method of lines (MOL), direct integration (DI) and the continued fraction (CF) method. In Sec. V we validate and comment our results comparing the QNM frequencies obtained via MOL, DI and CF methods. Furthermore, we investigate the static (marginally-stable) resonances of the polytropic hydrodynamic vortex, studying this system in the regime between stability and instability. We conclude with a brief discussion in Sec. VI.

II. THE POLYTROPIC HYDRODYNAMIC VORTEX
The (effective) spacetime of the polytropic hydrodynamic vortex is produced by an irrotational, barotropic and purely circulating fluid characterized by a polytropic equation of state. The line element of this system may be written as where ρ is the mass density, v θ is the angular component of the flow velocity v, i.e., v = v θθ , and c s is the speed of sound, which may be defined as assuming that the fluid is barotropic, i.e., where P is the hydrostatic pressure. Note that the quantities ρ, P , v and c s are given by the local properties of the unperturbed fluid flow [2]. We may obtain expressions for ρ, c s and v θ assuming that the fluid flow is irrotational and that it satisfies the Euler equation, i.e, Considering that density ρ, pressure P and angular flow velocity v θ are functions of the radial coordinate only, i.e., ρ = ρ(r), P = P (r), v θ = v θ (r), we obtain, respectively, from Eqs. (4) and (5), the following expressions where C is a constant related to the circulation of the fluid [13]. We may solve Eq. (8) using an expression that denotes the relation between P and ρ, i.e., an equation of state [cf. Eq. (3)]. Here we use a polytropic equation of state, namely where k p is the polytropic constant and N p is a constant called polytropic index [19]. Certain thermodynamical processes, in which one of quantities of the system remains constant, can be described by specific values of the polytropic index N p [19,20], namely: (i) isobaric (constant pressure) processes, described by polytropic index N p = −1; (ii) isometric (constant volume) and isopycnic (constant density) processes, described by polytropic index N p = 0; (iii) isothermal (constant temperature) processes, described by polytropic index N p = ±∞; and (iv) isentropic (constant entropy) processes, described by polytropic index N p = 1/(λ − 1). The quantity λ is the so-called specific heat ratio, being given by λ = c p /c v , where c p and c v are, respectively, the specific heat at constant pressure and the specific heat at constant volume, for a perfect gas [19].
Substituting Eq. (9) into Eq. (8), we obtain the following first-order differential equation: whose solution may be written as where ρ ∞ is the density at r → ∞. Note that the density goes to zero and becomes non-physical at a critical radius [21] defined by Here we defined K p ≡ k p ρ 1/Np ∞ . Therefore, the polytropic hydrodynamic vortex has an essential singularity at the critical radius r = r c , denoting that this spacetime, as well as Kerr spacetime, has a singularity with non-pointlike structure [18].
Using the equation of state given by Eq. (9) and the definition of the speed of sound given by Eq. (2), we may obtain an expression to the speed of sound, as a function of the density, for a polytropic fluid, i.e., Using Eqs. (11) and (13), we obtain get where c s∞ is the speed of sound at r → ∞, given by As it happens to the density [cf. Eq. (11)], the speed of sound [cf. Eq. (14)] becomes zero and non-physical at the critical radius r c [21] [see Eq. (12)]. The local Mach number is defined as being the ratio between absolute value of the flow velocity and speed of sound, i.e., M ≡ | v|/c s . Using the local Mach number as a parameter, we may define a supersonic (subsonic) flow as M > 1 (M < 1). For M = 1 (or, explicitly, | v| = c s ), we may write the outer boundary of the ergoregion, r e , as As seen from line element given by Eq. (1), the polytropic hydrodynamic vortex has no coordinate singularities, i.e., it has no event horizon, but it has an ergoregion with outer boundary at r = r e . Dividing Eq. (16) by Eq. (12) we find the following ratio between the radius of the outer boundary of the ergoregion r e and critical radius r c : From Eq. (17), for N p ≥ 0, we find that r e ≥ r c , i.e., the outer boundary of the ergoregion r e is located outside the outer boundary of the critical radius r c . For −1/2 ≤ N p < 0, the outer boundary of the ergoregion r e is located inside of the outer boundary of the critical radius r c , i.e., r e < r c . For N p < −1/2, the outer boundary of the ergoregion r e is complex, and therefore non-physical.
Basically, we are interested in values of the polytropic index such that r e is real and r e > r c , i.e., N p > 0. As seen in Eq. (12), the case in which N p = 0 gives us r c → 0, therefore we will not consider this case here. The N p = 0 case [which describe an isopycnic (constant density) processes] was analyzed in Ref. [16]. Here we focus our attention in the case of the polytropic index describing isentropic processes.
Essentially, an isentropic process is always an adiabatic process too, i.e., in addition to the entropy of the system remain constant, the process occurs without transfer of heat energy throughout [19]. The main motivation to study isentropic (adiabatic) processes here is based on fact that the propagation of sound waves in a gas is fundamentally an adiabatic process [22,23].
We now assume the polytropic hydrodynamic vortex to be composed by a perfect gas, described by the equation [19] where R = 8.314462 J K −1 mol −1 is the perfect gas constant, T is the temperature of the gas, and µ is the mean molecular weight of the gas. In this case, from Eqs. (9), (11) and (18), we may write expressions for the pressure and temperature of the gas, as functions of r, given, respectively, by where P ∞ is the pressure at r → ∞, and where T ∞ is the temperature at r → ∞. Pressure and temperature, as the density and speed of sound, becomes zero and non-physical at critical radius r c [21] (see Eq. (12)).
In Table I we exhibit estimates of the polytropic index N p , of the constant K p [which, using Eqs. (18) and (20), may be written as K p = RT ∞ /µ], of the critical radius r c , of the density at infinity ρ ∞ , of the speed of sound at infinity c s∞ , and of the ratio c s∞ /r c , for some gases.
In Table II we exhibit estimates of the density ρ(r), speed of sound c s (r), pressure P (r) and temperature T (r) for the Ethane gas (C 2 H 6 ) at different positions of the radial coordinate.
In Fig. 1 we plot, respectively, fluid density ρ, speed of sound c s , pressure P and temperature T , as functions of r and for different values of polytropic index N p .

III. PERTURBATIONS OF A POLYTROPIC HYDRODYNAMIC VORTEX
Linear perturbations propagating in acoustic spacetimes are described by the Klein-Gordon equation, namely where g ab is the contravariant effective metric, given by and g ≡ det(g ab ) = −ρ 4 r 2 /c 2 s . The spacetime of the polytropic hydrodynamic vortex is cylindrically symmetric, so that we may decompose the field Φ in terms of azimuthal harmonics, namely where m is an integer number that is related with the angular momentum of the perturbation. The coordinate In the frequency domain, we assume single-frequency modes ψ m (t, r) by using the following ansatz Substituting Eq. (24) into Eq. (23), we obtain Furthermore, we may use Eqs. (11), (12), (14), and (15) into Eq. (23), to obtain the following partial differential equation where we defined a dimensionless time coordinate τ , given by and a dimensionless radial coordinate x, namely x ≡ r/r c .
Equivalently, using Eqs. (11), (12), (14), and (15) into Eq. (25), we obtain the following ordinary differential equation for single-frequency modes u m (x) where we defined a dimensionless frequency , given by The ordinary differential equation given by Eq. (29) has regular singular points at the origin x = 0, at the critical radius x = x c ≡ 1, and at x = −1, and an irregular singular point at infinity (x → ∞). From Eq. (29), it is straightforward to see that the polytropic hydrodynamic vortex has the following symmetries in the frequency , related to azimuthal number m and circulation C where " * " denotes complex conjugation. Henceforth, taking into account these symmetries, we assume without loss of generality that m > 0 and C > 0.

A. Boundary and initial conditions
We consider two different boundary conditions at x min ≡ r min /r c , adopting essentially the same physically acceptable boundary conditions proposed in Ref. [16] (i.e., we consider a cylinder with radius r = r min made of a certain material with acoustic impedance Z), namely: (i) BC I, a boundary condition of Dirichlet type which mimics low-Z materials, and (ii) BC II, a boundary condition of Neumann type which mimics high-Z materials [25].
The boundary condition BC I may be defined as The boundary condition BC II is given by (cf. Eq. (22)) Both these boundary conditions are realistic and whether BC I or BC II hold in practice depends on how the experimental apparatus is implemented in the laboratory.
At large radial distances we require outgoing Sommerfeld or causal boundary conditions, which in the frequency domain, and given our choice of the Fourier transform, amount to As initial condition to time-domain analysis, we use the following Gaussian package and its first order derivative with respect to the time where x 0 is the position of the center of the peak of the Gaussian function (middle point), and σ sets the width of the Gaussian function. Note that the Gaussian package, given by Eq. (34), satisfies both boundary conditions BC I and BC II.
Our main task consists therefore in studying solutions of Eqs. (26) and (29), subjected to these boundary and initial conditions.

A. Method of lines (MOL)
We may determine the time-domain profiles associated with the QNMs of the polytropic hydrodynamic vortex applying a numerical method to solve the partial differential equation (26). We study the evolution of a Gaussian disturbance in the time domain, using the method of lines (MOL) as a numerical simulation [26,27]. The MOL involves a second-order spatial coordinate discretization and fourth-order Runge-Kutta method to advance in time [28][29][30].
To apply the MOL in Eq. (26), we discretize the radial coordinate x → x j = x min + jh (for a range x min ≤ x ≤ x max ), the wave function ψ m (τ, x) → ψ j , first-order and second-order spatial derivative ∂ 2 ∂x 2 ψ m (τ, x) → χ j , namely, where h is the grid spacing [26,30].
Employing the discretizations and transforming the spatial second-order derivative using the definition ζ j ≡ dψ j dτ , we may obtain from Eq. (26) a set of two first-order differential equations. Then, we apply the fourth-order Runge-Kutta method in each first-order differential equation to evolve the Gaussian perturbation (34) in time domain (for a range 0 ≤ τ ≤ τ max ) [28].

B. Direct integration (DI) method
In the frequency-domain, the differential equation (29) subjected to the boundary conditions described in Section III A is an eigenvalue problem for the frequency ω. Frequencies that satisfy the eigenvalue problem are called QNM frequencies. To investigate superradiant instabilities, we compute the QNM frequencies for the polytropic hydrodynamic vortex. QNMs are generically modes with complex frequencies. The time-dependence of the fluctuations, expressed in Eq. (24), implies that if the imaginary part of the QNM frequencies is negative (Im[ω] < 0) the spacetime is stable and the perturbations vanish for late times as ψ ∼ exp (−|Im[ω]|t). In contrast, if the imaginary part of the QNM frequencies is positive (Im[ω] > 0), the perturbations are amplified as ψ ∼ exp (Im[ω]t) and the spacetime is unstable.
We may solve directly the differential equation (29) using a direct integration (DI) method to obtain the QNM frequencies. The DI is based on the shooting method and numerical root-finding to obtain frequencies in the complex domain [31]. We write the outgoing solution at infinity as a generalized power series The series (38) and its first-order derivative are then used as boundary conditions to directly integrate Eq. (29) inwards for a range x max ≥ x ≥ x min . The QNM frequencies are the roots of this integration procedure that satisfy the appropriate BC I or BC II at x = x min , i.e., Eqs. (31) and (32), respectively. To find these roots, we use standard root-finding algorithms such as Newton's method.

C. Continued fraction (CF) method
An alternative to directly integrating Eq. (29), consists in expressing the problem as a continued fraction (CF) to find the QNM frequencies [32]. We may define the Frobenius-like series in the neighborhood of x = x min , namely Substituting Eq. (39) into Eq. (29), we find the following five-term recurrence relation: α 0 a 2 + β 0 a 1 + γ 0 a 0 = 0, (40a) α 1 a 3 + β 1 a 2 + γ 1 a 1 + δ 1 a 0 = 0, (40b) α n a n+2 + β n a n+1 + γ n a n + δ n a n−1 + n a n−2 = 0, where the recurrence coefficients are given by Using a double Gaussian elimination (cf. Refs. [16,33]) from the five-term recurrence relation (40) we may write the following three-term recurrence relation: α n a n+2 + β n a n+1 + γ n a n = 0, for n ≥ 0. (42) The recurrence coefficients α n , β n and γ n are complex functions that depend on the frequency , the azimuthal number m, the polytropic index N p and x min . For BC I, considering a 0 = 0 in Eq. (42), we obtain the following continued-fraction From Eq. (39) and considering n → n − 1 in Eq. (42) we find the following relation for BC II Eqs. (43) and (44) can be solved with standard rootfinding algorithms such as Newton's method.

V. RESULTS
Concerning the radial position r min where we impose boundary conditions BC I and BC II, the QNMs of the polytropic hydrodynamic vortex may be separated in two categories: (i) QNMs for r min located outside of the outer boundary of the ergoregion r e , which in dimensionless radial coordinate can be written as x min ≥ 2N p + 1 [see Eq. (17)].
(ii) QNMs for r min located inside of the outer boundary of the ergoregion r e , 1 < x min < 2N p + 1.
For r min located outside the outer boundary r e of the ergoregion, the polytropic hydrodynamic vortex admits only stable QNM frequencies; whereas for r min located inside r e , the polytropic hydrodynamic vortex admits both stable and unstable QNM frequencies [16]. Here we are especially interested in computing the QNM frequencies for r min located inside r e , which exhibit the ergoregion instabilities associated with this system.
We follow standard conventions of ordering the QNM frequencies (in dimensionless units) by their imaginary part [34]. The fundamental mode is the one with largest imaginary component Im[ ]. Thus, if the mode is unstable (Im[ ] > 0), the fundamental mode corresponds to the smallest instability timescale, and for stable (Im[ ] < 0) modes it corresponds to the longest-lived mode.
A. Boundary conditions imposed outside of the outer boundary of the ergoregion We have computed QNM frequencies for r min located outside of the outer boundary r e of the ergoregion, checking the agreement between three methods for different values of the polytropic index N p . Examples are shown in Table III.
In Table III we exhibit the estimates of the QNM frequencies obtained via MOL, DI and CF methods for different values polytropic index N p . We have obtained excellent agreement between three methods.  Table I Next, we choose r min located inside of the outer boundary r e of the ergoregion and outside of the critical radius r c (1 < x min < 2N p + 1).
In Table IV we exhibit the estimates of the QNM frequencies for azimuthal number m = 5, obtained via MOL, DI and CF methods, for different values of polytropic index N p .  In Fig. 2 we plot real and imaginary parts of the fundamental (n = 0) QNM frequencies , for different values of azimuthal numbers m and polytropic index N p = 4.54, obtained via CF method, imposing boundary conditions BC I and BC II at different values of x min . Analyzing the imaginary part of the QNM frequencies , we conclude that, as the azimuthal number m increases, the threshold between stability and instability (i.e., in the neighborhood of Im( ) = 0) increases, tending to the outer boundary r e of the ergoregion, which in dimensionless radial coordinate may be represented by x min = x e (where x e = r e /r c ). This behavior can be seen more clearly in the zooms for BC I and BC II exhibited in the bottom plots of Fig. 2 , being most prominent to boundary conditions BC II than BC I.
As an example of a possible experimental implementation, we estimate the QNM frequencies ω for a vortex made of Ethane gas (whose polytropic constant K p = 7.96745 × 10 4 m 2 /s 2 and polytropic index N p = 4.54). We may obtain the QNM frequencies ω using Eq. (30), and the data exhibited in Tables I, II and IV . To this particular experimental setup, we consider the circulation C = 0.5 m 2 /s, subject to a perturbation with azimuthal number m = 5 and we impose boundary conditions of Neumann type (BC II) at r min = 2.0r c . The real and imaginary parts of the fundamental (n = 0) QNM frequency ω are then, respectively, Re(ω) = 0.304104 MHz and Im(ω) = 0.410177 s −1 . From the imaginary part it is possible to estimate the instability timescale of this perturbation, which is t scale ≡ 1/Im(ω) = 2.43797 s.

D. Static (marginally-stable) resonances
Now, we investigate a critical configuration of marginal-stability of the polytropic hydrodynamic vortex, referred to as static (marginally-stable) resonances [35,36], mathematically represented by Re (ω) = Im (ω) = 0. (It may be seen in Fig. 2 that for specific values of x min the QNMs frequencies goes to zero.) An analytical study of a particular case of the incompressible hydrodynamic vortex (N p = 0), in its marginally-stable regime, was addressed in Ref. [35]. Here we study a compressible system characterized by the polytropic equation of state (8).
Considering ω = 0, we may obtain the following analytical solution for the ordinary differential equation (29): where a 1 and a 2 are integration constants, a ± = i 7/2∓m x 1/2±m , with 2 F 1 [α, β, γ, y] being the hypergeometric function [37] and b = N 2 p + m 2 (1 + 2N p ). We assume that a 1 = 0, so that the physically acceptable solution (45) is finite at x → ∞. It follows that Applying the boundary conditions, BC I [given by Eq. (31)] and BC II [given by Eq. (32)] in the solution given by Eq. (47), we may write the following equations respectively, where x (n) min is the nth positive root of Eqs. (48) (for BC I) and (49) (for BC II) and describe the static (marginally-stable) resonances of the polytropic hydrodynamic vortex. To find these roots, we use a standard root-finding algorithms such as Newton's method.
In Table V we exhibit the estimates of x  In Fig. 3 we plot the estimates ofx ≡ r   x min  Table I).

VI. CONCLUSION
Here we used the method of lines (MOL), direct integration (DI) and continued-fraction (CF) methods to obtain the QNM frequencies of the polytropic hydrody-namic vortex, an effective spacetime produced by purely circulating compressible fluid that satisfies a polytropic equation of state. To validate our results, we compared the QNM frequencies obtained via these three different methods and obtained excellent agreement. Further- more, we studied the polytropic hydrodynamic vortex in the regime between stability and instability, obtaining the configuration of the parameters of this system (azimuthal number m and polytropic index N P ) required for the onset of the ergoregion instability. We focused our attention on the dependence of the QNM frequencies with the polytropic index N p . As the polytropic index N p increases, the magnitude of real and imaginary parts of the QNM frequencies increase (decrease) for unstable (stable) modes. Furthermore, we have shown that the polytropic hydrodynamic vortex, a compressible system with an ergoregion and without an event horizon is unstable. Together with the instability of the corresponding incompressible system, namely, the hydrodynamic vortex studied in Ref. [16], this establishes the ergoregion instability as a generic phenomena. The instability is more clearly revealed by computing QNMs, imposing boundary conditions at r = r min located inside the outer boundary r e of the ergoregion. Finally, we have shown that the onset of ergoregion instability approaches the outer boundary of the ergoregion as the azimuthal number m increases, denoting that for large values of the azimuthal number m the QNMs become unstable, independently of the configuration (circulation C and polytropic index N p ) of the system (as may be seen in Fig. 2 ). Furthermore, we have shown that for large values of azimuthal number