On the Viscous Ringed Disk Evolution in the Kerr Black Hole Spacetime

Abstract

Supermassive black holes (SMBHs) are observed in active galactic nuclei interacting with their environments, where chaotical, discontinuous accretion episodes may leave matter remnants orbiting the central attractor in the form of sequences of orbiting toroidal structures, with strongly different features as different rotation orientations with respect to the central Kerr BH. Such ringed structures can be characterized by peculiar internal dynamics, where co-rotating and counter-rotating accretion stages can be mixed and distinguished by tori interaction, drying–feeding processes, screening effects, and inter-disk jet emission. A ringed accretion disk (RAD) is a full general relativistic model of a cluster of toroidal disks, an aggregate of axi-symmetric co-rotating and counter-rotating disks orbiting in the equatorial plane of a single central Kerr SMBH. In this work, we discuss the time evolution of a ringed disk. Our analysis is a detailed numerical study of the evolving RAD properties formed by relativistic thin disks, using a thin disk model and solving a diffusion-like evolution equation for an RAD in the Kerr spacetime, adopting an initial wavy (ringed) density profile. The RAD reaches a single-disk phase, building accretion to the inner edge regulated by the inner edge boundary conditions. The mass flux, the radial drift, and the disk mass of the ringed disk are evaluated and compared to each of its disk components. During early inter-disk interaction, the ring components spread, destroying the internal ringed structure and quickly forming a single disk with timescales governed by ring viscosity prescriptions. Different viscosities and boundary conditions have been tested. We propose that a system of viscously spreading accretion rings can originate as a product of tidal disruption of a multiple stellar system that comes too close to an SMBH.

1. Introduction

In this analysis, we consider a cluster of general relativistic viscous disks orbiting a central Kerr black hole (BH). We study the (diffusive) evolution of the disks agglomerate following the approach developed in [1] for a single general relativistic thin disk orbiting a Kerr BH, applied here to a general relativistic disk characterized by an internal ringed structure composed of different co-rotating and counter-rotating ring-like components (with respect to the central spinning BH).

From the point of view of numerical integration, we address the issue of the condition at the inner edge, focusing on the issue of a single (geometrically thin) disk. We examine ring aggregates with the same relative rotation orientation (ℓco-rotating), composed of co-rotating or counter-rotating rings with respect to the central BH, and clusters of rings with opposite relative rotation orientation (ℓcounter-rotating), constituted by an inner co-rotating ring and an outer counter-rotating ring, or vice versa, by an inner counter-rotating ring and an outer co-rotating ring with respect to the central BH. Combined sets of clusters are then considered. We will evaluate the evolution of the fluid density radial drift, the disk mass $M_D$, and the mass flux as dependent on the BH spin, the tori rotation orientation, and the initial density profiles and spreading. We proceed by adopting an adapted initial condition on the ring’s initial density inhomogeneity and spreading, and we investigate the solutions of the diffusion equations for one system with an initial wavy (ringed) density profile, modeling the two ℓco-rotating rings with the same viscosity prescription. In order to do this, we shall make use of the ringed accretion disk (RAD) model for aggregates of toroidal axis-symmetric disks of co-rotating and counter-rotating fluid structures (tori) orbiting one center supermassive Kerr BH, with a symmetry plane coinciding with the equatorial plane of the central Kerr BH [2]. The advantage is in the fact that the model strongly binds the disks characteristic to the BH defining parameters. The RAD frame constrains the initial density profile for aggregates of co-rotating and counter-rotating rings. We use different viscosity prescriptions for tori with different rotation orientations with respect to the central attractor. Doublet, triplet, and quadruplet rings with different relative rotations are also explored.

Aggregates of orbiting toroids can form with strongly different features as different rotation orientations with respect to the central Kerr BH, for example, in active galactive nuclei, where disks, forming by several matter remnants around a single central attractor, may contain traces of chaotical accretion episodes.

We should note that the accretion rate can be modified by the interaction between disks, combined with the multi-phase accretion processes of the different components of the cluster. This fact could constitute a mechanism to increase the mass accretion rates and the mass of the supermassive BHs. Multi tori orbiting a single attractor can host screening tori. This could be the case, for example, in the double system constituted by an internal co-rotating ring, accompanied by an external counter-rotating toroid that is accreting into the central BH. Therefore, this effect would frame multiple and simultaneous accretion phases, including inter-disk accretion and jet shell emission. Observational evidence of these phenomena might be found in a radially stratified emission of the X-ray spectrum.

As the members of a multiple stellar system become tidally disrupted, accretion rings can be produced. We envisage a system of such viscously spreading rings as a model to describe the subsequent evolution and interaction of the debris. In this context, it is interesting to note that, very recently, Peissker et al. (2024) reported on the discovery of a binary system within the S-cluster close to the supermassive BH (SMBH of mass about $4\times {10}^6{M}_{\odot }$). In fact, the central parsec of the Milky Way’s SMBH (Sagittarius A*) contains a very dense population of stars: the nuclear star cluster, one of the densest and most massive stellar systems studied so far. Thanks to its relative proximity at a distance of about 8 kpc from us, detailed kinematics of individual stars can be achieved, although the measurements are hampered by obscuration effects along the line of sight and confusion among fast-moving stars. Similar systems can be expected in the nuclei of other quiescent galaxies. Let us emphasize that the galactic center is an extreme example of an inactive nucleus (the accretion rate is very low, no accretion disk is present, and the accretion activity is suppressed); in this case, the signal tidal disruption of stars is expected to be very clean [3].

We first address the case of ℓco-rotating tori, the composition of two independent evolutions, and ℓcounter-rotating rings with different viscosity profiles. The impact on the cluster evolution of the initial density profiles, including the initial ringed components’ differences in density and spreading, is evaluated. We expect that within this approximation, due to the combination of ℓcounter-rotating seeds, the evolution observed could be qualitatively similar to ℓco-rotating seed evolution.

The evolution of a geometrically thin disk subjected to a viscous torque was examined in [4,5] with a diffusion-like equation (see also [6,7]). The spreading of a viscous disk in the relativistic regime was also developed by [8,9,10,11,12,13] (see also [14,15,16]). Relativistic equations (one-dimensional relativistic equations) reduce to the Keplerian equations in the Newtonian limit. In [17], the cluster diffusive evolution was analyzed following [8,9,10,11,12,13], describing the evolution of the azimuthal averaged and height integrated disk surface density governed by different viscosity prescriptions and different inner boundary conditions. The case of BH attractors with spin $a=\pm 0.2$ and with spin $a=\pm 0.9$ were investigated and compared with the limiting case of Schwarzschild static attractors. In this context, the disk’s total luminosity, flux, and mass rate were also examined, and it was proved that the disk’s luminosity curves present a fingerprint of the initial inner ringed structure of the disk and at different times.

The layout of this paper is as follows. The spacetime metric and constants of motion are introduced in Section 2. Some notes on the ringed accretion disks are presented in Section 2.1.

The evolution of viscous tori is studied in Section 3 following [1], where the dynamics of a viscous gas ring (single thin-disk approximation) orbiting a central Kerr BH is discussed. The surface density equation is solved using a perturbation approach adapted to the thin-disk approximation with a low-viscosity assumption. As in [4,5], the general relativistic equation for surface density dynamics is found from the conservation of the energy-momentum tensor for a viscose fluid, including a radiative energy flux and its projection along the flow velocity $U^a$. Different values of constant viscosity are assumed. RADs orbiting a central attractor with spin $a=\pm 0.9$ are explored in Section 3.1. In Section 3.2, RADs orbiting a slower spinning BH with spin $a=\pm 0.2$ are investigated. The mass flux, the total mass of the disk, and the radial drift of maximum densities in the RAD are also studied.

Concluding remarks follow in Section 4. In Appendix A, there are some notes on the conditions of the null flux $\mathcal{F}$ in the frame explored in Section 3.

2. The Spacetime Metric and Constants of Motion

In the Boyer–Lindquist (BL) coordinates $\{t,r,\theta ,\varphi \}$1, the Kerr spacetime metric reads

$$\begin{array}{c}d{s}^2=-\left(1-{\displaystyle \frac{2Mr}{\Sigma }}\right)d{t}^2+{\displaystyle \frac{\Sigma }{\Delta }}d{r}^2+\Sigma d{\theta }^2+\left[(r^2+a^2)+{\displaystyle \frac{2Mr{a}^2}{\Sigma }}{sin}^2\theta \right]{sin}^2\theta d{\varphi }^2\hfill \\ -{\displaystyle \frac{4rMa}{\Sigma }}{sin}^2\theta dtd\varphi ,\hfill \end{array}$$

(1)

$$\begin{array}{c}\Delta \equiv a^2+r^2-2rM\mathrm{and}\Sigma \equiv a^2(1-{sin}^2\theta )+r^2.\hfill \end{array}$$

(2)

The equatorial plane, $\sigma \equiv {sin}^2\theta =1$, is a metric symmetry plane coincident with the disk’s symmetry plane. Parameter $a=J/M\ge 0$ is the metric spin, where the total angular momentum is J and the gravitational mass parameter is M. A Kerr BH is defined by the condition $a\in [0,M]$ with killing horizons $r_{-}\le r_{+}$, where $r_{\pm }\equiv M\pm \sqrt{M^2-a^2}$. An extreme Kerr BH has dimensionless spin $a/M=1$, and the non-rotating case $a=0$ is the Schwarzschild BH solution. The spacetime outer ergoregion is $]r_{+},r_{ϵ}^{+}]$, where the outer and inner stationary limits $r_{ϵ}^{\pm }$ (ergosurfaces) are given by $r_{ϵ}^{\pm }\equiv M\pm \sqrt{M^2-a^2(1-\sigma )}$, respectively, where $r_{ϵ}^{-}=0$ and $r_{ϵ}^{+}=2M$ in the equatorial plane $\theta =\pi /2$ ($\sigma =1$), and $r^{+}<r_{ϵ}^{+}$ on $\theta \ne 0$.

(In the following, we will use geometrical units where appropriate to make the reading of dimensionless variables easy, where $r\to r/M$ and $a\to a/M$).

We introduce the constants of motion $(\mathcal{E},\mathcal{L})$ related to the killing field ${\xi }_t={\partial }_t$ and ${\xi }_{\varphi }={\partial }_{\varphi }$, respectively, with

$$\begin{array}{c}\mathcal{E}=-(g_{t\varphi }\dot{\varphi }+g_{tt}\dot{t}),\mathcal{L}=g_{\varphi \varphi }\dot{\varphi }+g_{t\varphi }\dot{t},g_{ab}{U}^a{U}^b=\kappa {\mu }^2.\hfill \end{array}$$

(3)

$U^a\equiv \{\dot{t},\dot{r},\dot{\theta },\dot{\varphi }\}$. The notation $\dot{q}$ is for the derivative of q with respect to the proper time for $\mu >0$, and $\kappa =-1$ is a normalization constant.

The velocity components $(U^t,U^{\varphi })$ are

$$\begin{array}{c}\hfill U^t={\displaystyle \frac{g_{\varphi \varphi }\mathcal{E}+g_{t\varphi }\mathcal{L}}{g_{t\varphi }^2-g_{\varphi \varphi }{g}_{tt}}},U^{\varphi }=-{\displaystyle \frac{g_{t\varphi }\mathcal{E}+g_{tt}\mathcal{L}}{g_{t\varphi }^2-g_{\varphi \varphi }{g}_{tt}}},\end{array}$$

(4)

where these components also describe the circular orbits in BL coordinates. We also introduce the specific angular momentum ℓ and the relativistic angular velocity $\Omega \equiv U^{\varphi }/U^t$, where

$$\begin{array}{c}\ell \equiv {\displaystyle \frac{\mathcal{L}}{\mathcal{E}}}=-{\displaystyle \frac{g_{\varphi \varphi }{U}^{\varphi }+g_{\varphi t}{U}^t}{g_{tt}{U}^t+g_{\varphi t}{U}^{\varphi }}}=-{\displaystyle \frac{g_{t\varphi }+g_{\varphi \varphi }\Omega }{g_{tt}+g_{t\varphi }\Omega }}.\hfill \end{array}$$

(5)

2.1. Ringed Accretion Disks

Ringed accretion disks (RADs) model a cluster of general relativistic toroidal disks orbiting a central Kerr BH. In this analysis, we consider the disks sharing the equatorial plane of the Kerr central attractor and the RAD being composed by a mixing of co-rotating and counter-rotating rings. Many properties of the aggregate are constrained by the geometry of the Kerr spacetime and, in particular, by the equatorial geodesic structure. This is constituted by the marginally stable circular orbit, $r_{mso}^{\pm }$, the marginally bounded orbit, $r_{mbo}^{\pm }$, and the marginally circular orbit (photon circular orbit), $r_{\gamma }^{\pm }$. Here and in the following, we use the notation $(\pm )$ for counter-rotating and co-rotating motion, respectively.

A free-falling condition for the accretion from the disk inner edge $r_{inner}\in [r_{mbo}^{\pm },r_{mso}^{\pm }]$ is assumed in many approximations. The assumption that matter in accretion follows, for $r<r_{inner}$, a geodesic motion, having a non-null radial velocity component $U^r\ne 0$, grounds the single ring perturbation for the diffusive equation, and here, we extend this approach to the RAD analysis.

Constraints from RAD Systems

Here, we summarize the main features of the ringed accretion disk (RAD) systems, setting the constraints for the analysis of the diffusive equations of ring clusters orbiting in Kerr spacetimes.

The ringed accretion disks were studied in [2]. RAD emerges as a single, geometrically thin accretion disk characterized by an internal ringed structure [2].

To simplify our discussion, below we describe each toroid of the ringed disk as a geometrically thick torus governed by the Polish donut (P-D) model. Nevertheless, RAD construction is independent of the adopted model for the ring constituent.

Polish donut (P-D)

In geometrically thick Polish donut (P-D) tori models, the Euler equation describes the axially symmetric stationary general relativity hydrodynamic (GRHD) perfect fluid barotropic tori, governing the force balance in the disk between centrifugal, pressure, and gravitational forces2.

These are geometrically thick, optical opaque, general relativistic barotropic axially symmetric and stationary toroidal configurations, cooled by advection and composed by a one particle-specie perfect fluid, where ${\partial }_t\mathrm{Q}=0$ and ${\partial }_{\varphi }\mathrm{Q}=0$, with $\mathrm{Q}$ being a generic spacetime tensor. The continuity equation is identically satisfied as a consequence of the symmetries, and the fluid dynamics are governed by the Euler equation only, assuming a barotropic equation of state and fluid orbital motion with $U^{\theta }=0$ and $U^r=0$.

Toroidal surfaces are closed quiescent ($\mathrm{C}$) (without cusp), proto-jets ${\mathrm{O}}_{\times }$, and closed-cusped $\left({\mathrm{C}}_{\times }\right)$ solutions (proto-jets are open-cusped solutions associated with geometrically thick disks. They are characterized by matter funnels along the direction of the BH rotational axis).

In the first and simplest model, a constant fluid-specific angular momentum is assumed ℓ [2,19]; by setting the ℓ = constant as a parameter, the maximum density points in the disk are fixed. More specifically, the pressure gradients (defined by the Euler equations) are regulated by the gradients of an effective potential function for the fluid $V_{eff}(r;\ell ,a)$ parameterized with a constant specific angular momentum $\ell \equiv \mathcal{L}/\mathcal{E}$.

The force balance in the disk is regulated by an effective potential function and encoded by the Boyer condition of equilibrium configurations of rotating perfect fluids. According to Boyer’s theory on equipressure surfaces applied to a P-D torus, toroidal surfaces are equipotential surfaces with effective potential.

The minima of the effective potential are the maxima of pressure in the torus (torus center $r_{center}$), and the minima of pressure are the surfaces’ cusp $r_{\times }$ for closed-cusped tori and $r_j$ for proto-jets (see [19]).

The location of pressure extremes is constrained by the spacetime (extended) geodesic structure constituted by radii $\{r_{mso}^{\pm },r_{\gamma }^{\pm },r_{mbo}^{\pm }\}$, the marginally stable and marginally circular (photon) orbit, and the marginally bounded orbits for ${\mathit{\ell }}^{\pm }$ and radii $\{r_{\left[mbo\right]}^{\pm },r_{\left[\gamma \right]}^{\pm }\}$ together, as shown in Figure 1 and defined in

Table 1. Co-rotating and counter-rotating tori in Kerr BH spacetimes. ${\ell }^{\pm }$ is the fluid specific angular momentum for counter-rotating $(+)$ and co-rotating $(-)$ fluids, respectively. We adopt the notation $q_{•}\equiv q\left(r_{•}\right)$ for any quantity q evaluated on a radius $r_{•}$. Radii $\{r_{mso}^{\pm },r_{\gamma }^{\pm },r_{mbo}^{\pm }\}$ are marginally stable and marginally circular (photon) orbits and the marginally bounded orbits for ${\ell }^{\pm }$ specific angular momentum. Radii $\{r_{\left[mbo\right]}^{\pm },r_{\left[\gamma \right]}^{\pm }\}$, together with $\{r_{mso}^{\pm },r_{mbo}^{\pm },r_{\gamma }^{\pm }\}$, constitute the Kerr geometry extended geodesic structure. $r_{\times }$ ($r_j$) is the torus (proto-jets) cusp (“inner edge” minimum pressure and density points in the P-D model) and $r_{center}$ is the torus center (maximum pressure and density points in the P-D model) (see Figure 1).

$r_{\left[mbo\right]}^{\pm }:{\ell }^{\pm }\left(r_{mbo}^{\pm }\right)={\ell }^{\pm }\left(r_{\left[mbo\right]}^{\pm }\right)\equiv {\ell }_{mbo}^{\pm },r_{\left[\gamma \right]}^{\pm }:{\ell }^{\pm }\left(r_{\gamma }^{\pm }\right)={\ell }^{+}\left(r_{\left[\gamma \right]}^{+}\right)\equiv {\ell }_{\gamma }^{+}$

$r_{\gamma }^{\pm }<r_{mbo}^{\pm }<r_{mso}^{\pm }<r_{\left[mbo\right]}^{\pm }<r_{\left[\gamma \right]}^{\pm }$

${\ell }^{\pm }\in {\mathbf{L}}_{\mathbf{1}}^{\pm }\equiv ]{\ell }_{mbo}^{\pm },{\ell }_{mso}^{\pm }[$: quiescent and cusped tori. $r_{\times }\in ]r_{mbo}^{\pm },r_{mso}^{\pm }]$, $r_{center}\in ]r_{mso}^{\pm },r_{\left[mbo\right]}^{\pm }]$;

${\ell }^{\pm }\in {\mathbf{L}}_{\mathbf{2}}^{\pm }\equiv [{\ell }_{\gamma }^{\pm },{\ell }_{\mathrm{mbo}}^{\pm }[$: quiescent tori and proto-jets. $r_j\in ]r_{\gamma }^{\pm },r_{mbo}^{\pm }]$, $r_{center}\in ]r_{\left[mbo\right]}^{\pm },r_{\left[\gamma \right]}^{\pm }]$;

${\ell }^{\pm }\in {\mathbf{L}}_{\mathbf{3}}^{\pm }:{\ell }^{\pm }<{\ell }_{\gamma }^{\pm }$: quiescent tori, $r_{center}>r_{\left[\gamma \right]}^{\pm }$.

.

The specific angular momentum ${\ell }^{\pm }(r;a)\equiv {\mathcal{L}}^{\pm }/{\mathcal{E}}^{\pm }$ distribution on the equatorial plane for fixed spacetime spin a contains the distributions of all the possible tori centers, $r_{center}^{\pm }>r_{mso}^{\pm }$, and cusps, $r_{inner}^{\pm }\in ]r_{\gamma }^{\pm },r_{mso}^{\pm }[$, where $r_{inner}^{\pm }\in \{r_j^{\pm },r_{\times }^{\pm }\}$ for counter-rotating and co-rotating tori, respectively, as shown in Table 1.

The P-D models allow an immediate, simplistic, limiting concept of the inner/outer edge of both quiescent and cusped torii, being defined and identified uniquely by the torus equipotential surface (for quiescent torii) or also by the only angular momentum parameter ℓ, which are constant for each torus for cusped surfaces, governed by the geodesic structure.

These features constitute a model with great advantages and render their adoption extremely useful and predictive; therefore, P-D tori are well known and widely used in the literature also as the initial conditions of the more complex evolution of GRMHD (magnetohydrodynamic) accretion structures (see, for example, ref. [20]).

Clusters constraints

An RAD seed (RAD of the order $n=2$) is a couple of tori. We can concentrate our attention on two tori with parameters $({\ell }_i,{\ell }_o)$ for the inner and outer tori, respectively, with respect to the central BH. Introducing notation ≶, there is ${\mathrm{C}}_i<{\mathrm{C}}_o$ for the ring’s relative location (relative location of the tori centers). By construction, the initial data on the ring’s seeds describe separated tori. Hence, we have

$$\begin{array}{c}\hfill r_{inner}\left(i\right)<r_{center}\left(i\right)<r_{outer}\left(i\right)<r_{inner}\left(o\right)<r_{center}\left(o\right)<r_{outer}\left(o\right),\end{array}$$

(6)

where $r_{outer}$ is the ring’s outer edge (with the inner edge $r_{\mathrm{IN}}\in ]r_{mbo}^{\pm },r_{mso}^{\pm }]$).

In the RAD seeds, we need to introduce the concept of ℓco-rotating tori, defined by the condition ${\ell }_i{\ell }_o>0$, and ℓcounter-rotating tori, defined by the relation ${\ell }_i{\ell }_o<0$. The two ℓco-rotating tori can both be co-rotating ($\ell a>0$) or counter-rotating ($\ell a<0$) with respect to the central attractor.

We can introduce the spins $\{a_1,a_2,a_3\}$, defined by the crossing of radii of the spacetime geodesic structure (shown in Figure 1 and defined in Table 1), as follows: $a_1\equiv 0.4740M:r_{\left[\mathrm{mbo}\right]}^{+}\left(a_1\right)=r_{\left[\gamma \right]}^{-}\left(a_1\right)$, $a_2=0.461854M:r_{\left[\mathrm{mbo}\right]}^{-}\left(a_2\right)=r_{\mathrm{mso}}^{+}\left(a_2\right)$ and $a_3\equiv 0.73688M:r_{\left[\gamma \right]}^{-}\left(a_3\right)=r_{\mathrm{mso}}^{+}\left(a_3\right)$. Spins $\{a_1,a_2,a_3\}$ identify classes of attractors characterized by the different relative locations of the radii of the geodesic structure. Focusing on a seed with cusped tori, the RAD of the order $n=2$ can be classified into four classes. The first class of seeds is constituted by the tori ${\mathrm{C}}_{\times }^{\pm }<{\mathrm{C}}^{\pm }$, which are an ℓco-rotating seed of counter-rotating or co-rotating rings with an inner cusped disk. The seed can be observed in general orbiting any Kerr BH attractor. This case also includes any seeds orbiting a Schwarzchild BH. The second case is the pair ${\mathrm{C}}_{\times }^{+}<{\mathrm{C}}^{\pm }$. We can generalize this case to the pair ${\left(\right)}^{+}<{\mathrm{C}}^{-}$, including an inner accreting $\left({\mathrm{C}}_{\times }\right)$ or non-accreting $\left(\mathrm{C}\right)$ torus, indicated here with notation $\left(\right)$. These pairs can be observed in all spacetimes with $a\in [0,M]$. For $a⪆a_3$, the outer torus should be very far from the inner torus to prevent collision, i.e., with $r_{center}>r_{\left[\gamma \right]}^{-}$. However, in spacetimes with $a\in [0,a_2[$ only, the quiescent co-rotating torus ${\mathrm{C}}^{-}$ approaches the cusped phase, i.e., $r_{\times }⪆r_{\mathrm{mso}}^{-}$, in the RAD seed. The third class is the pair ${\mathrm{C}}_{\times }^{-}<{\mathrm{C}}^{\pm }$, which can orbit in all spacetimes with $a\in [0,M]$. Finally, the fourth case is constituted by ℓcounter-rotating seeds ${\mathrm{C}}_{\times }^{-}<{\mathrm{C}}_{\times }^{+}$, featuring double accreting disks, formed by an inner co-rotating cusped disk and an outer counter-rotating cusped disk. This RAD can be observed in all Kerr geometries. Whereas, the slower the BH ($a⪅a_1$), the lower the specific angular momentum ${\ell }^{-}$ of the inner co-rotating torus, and the smaller the tori spacings.

It is worth noting that if the accreting torus is counter-rotating with respect to the Kerr attractor, i.e., a ${\mathrm{C}}_{\times }^{+}$, then there is no inner counter-rotating torus, but a couple may be formed as a ${\mathrm{C}}_{\times }^{+}<{\mathrm{C}}^{\pm }$ or as a ${\left(\right)}^{-}<{\mathrm{C}}_{\times }^{+}$ one.

These configurations are determined by the constraints placed on the inner edge (and in some cases on the torus center and outer edge), which are bounded in the range $]r_{mbo}^{\pm },r_{mso}^{\pm }]$ (a constraint shared by most accretion disk models).

3. Evolution of Viscous Tori

In [1], the dynamics of a viscous gas ring orbiting a central Kerr BH were discussed within the (single) thin-disk approximation. The surface density equation was solved considering the marginally stable orbit as the disk’s inner edge, with boundary conditions for the accretion disk model and its time evolution. Following [1], the Kerr background metric is set in Boyer–Lindquist cylindrical coordinates. Using a perturbation approach (the disk height is much smaller than the radial length scale) with a low-viscosity assumption, as in the Keplerian frame, the GR equations for the disk dynamics can be recovered by the conservation of the energy-momentum tensor for a viscose fluid, including a radiative energy flux and its projection along the flow velocity $U^a$, i.e.,

$$\begin{array}{c}{\nabla }_a{T}^{ba}=0,{\nabla }_a\left(\rho U^a\right)=0,\mathrm{where}{T}^{ab}=(\rho +ϵ+p)U^a{U}^b+p{g}^{ab}+S^{ab}+{\tau }^{ab},\hfill \\ \mathrm{with}{\tau }^{ab}\equiv U^a{q}^b+q^a{U}^b,\mathrm{and}{U}_a{S}^{ab}=0,U_a{q}^a=0,\hfill \end{array}$$

(7)

where ${\tau }^{ab}$ can be considered a radiation stress term, $\rho$ is the rest mass density of matter, p the pressure, and $ϵ$ the internal energy. The viscous stress tensor $S^{ab}$ encodes the viscous contribution, and $q^a$ is an energy flux term (radiative energy flow vector)3. We also assume

$$\begin{array}{c}\hfill S^{ab}=-\nu \rho t^{ab}\mathrm{with}{t}^{ab}\equiv h^{ac}{\nabla }_c{U}^b+h^{bc}{\nabla }_c{U}^a-{\displaystyle \frac{2}{3}}{\nabla }_c{U}^c{h}^{ab},\end{array}$$

(8)

where the viscosity is $\nu =\nu \left(r\right)$, $t^{ab}$ is the trace-free shear tensor, and $h^{ab}\equiv g^{ab}+U^a{U}^b$ is the projector of the three-dimensional subspace orthogonal onto $U^a$.

Similarly to RAD, it is assumed that ${\partial }_{\varphi }Q=0$ for any matter quantity (axially symmetric disk). Moreover, we assume that the rest mass is predominant over the pressure energy density and shear stress. However, in the thin disk approximation, all quantities are independent from z, which is therefore ruled out by vertical integration, and we consider the (vertically integrated) surface density $\Sigma$, replacing $\rho$ in the continuity equation4. Here, the disks are in circular orbits. We assume the condition $U^r=0$. Therefore, without viscosity, there would be no radial inflow. Viscosity causes a small perturbation, and we can assume, for the particles gas, a geodesic (Keplerian) motion with velocities ${\tilde{U}}^a$. These will be deviated by the quantities ${\xi }^a$, viscosity terms which are null for $\nu \to 0$. We will only consider terms in first-order in ${\xi }^a$ (note that $\nu {\xi }^a$ is a second-term-order perturbation).

Using the approximations in the momentum balance equation, we obtain ${\tilde{t}}^{ab}$, the shear expression, considering the geodesic four-velocity and the constraint ${\tilde{U}}_t{\tilde{t}}^{tr}+{\tilde{U}}_{\varphi }{\tilde{t}}^{\varphi r}=0$, where there is

$$\begin{array}{c}\hfill {\tilde{t}}_{\mp }^{\varphi r}=\mp {\displaystyle \frac{3\sqrt{\frac{M}{r}}{\mathcal{T}}_{*}^{\pm }\Delta }{2r^3{\left(S_{*}^{\pm }\right)}^{3/2}}}\mathrm{where}{S}_{*}^{\pm }\equiv r^2\pm 2a\sqrt{Mr}-3Mr,{\mathcal{T}}_{*}^{\pm }\equiv r^2\pm a\sqrt{Mr}-2Mr\end{array}$$

(9)

for co-rotating $\left({\tilde{t}}_{-}^{\varphi r}\right)$ and counter-rotating $\left({\tilde{t}}_{+}^{\varphi r}\right)$ fluids, respectively.

Introducing the following quantities,

$$\begin{array}{c}{A}_{-}\equiv {\displaystyle \frac{2\sqrt{\frac{r}{M}}{S}_{*}^{+}}{r^2-3a^2+8a\sqrt{Mr}-6Mr}},B_{-}\equiv {\displaystyle \frac{3\sqrt{Mr}{\Delta }^2}{2r^2{S}_{*}^{+}}},C_{-}\equiv {\displaystyle \frac{\sqrt{S_{*}^{+}}}{r^2+a\sqrt{Mr}}},\hfill \end{array}$$

(10)

where the mass flux is

$$\begin{array}{c}{\mathcal{F}}_{\mp }\equiv r\Sigma {\xi }^r=-A_{\pm }{\partial }_r(\nu B_{\pm }\Sigma ).\hfill \end{array}$$

(11)

Here, there is $C_{+}=C_{-}(a\to -a)$, $B_{+}=-B_{-}(a\to -a)$, and $A_{+}=-A_{-}(a\to -a)$, and ${\mathcal{F}}_{-}\left(a\right)={\mathcal{F}}_{+}(-a)=\mathcal{F}\left(a\right)$). Analogously, using the continuity equation, we obtain the equation for the density evolution as follows:

$$\begin{array}{c}{\partial }_t{\Sigma }_{\pm }=C_{\pm }{\partial }_r\left[A_{\pm }{\partial }_r(\nu B_{\pm }\Sigma )\right]\hfill \end{array}$$

(12)

for counter-rotating and co-rotating fluids, respectively5. Then, considering Equation (12), ${\partial }_t{\Sigma }_{-}={\partial }_t{\Sigma }_{+}(a\to -a)$ and ${\partial }_t{\Sigma }_{-}(a=0)={\partial }_t{\Sigma }_{+}(a=0)$. This setup provides the time evolution of a ring under the combined effects of viscous torques and BH gravity.

We integrate the evolution equations in Equation (12) by first considering ℓco-rotating rings and then combining the sequence sin ℓcounter-rotating disks sequences. For two BH spins, using dimensionless units, we assume a constant viscosity6$\nu$. The equations singularities are the orbits $r_{\gamma }^{\pm }$ and $r_{mso}^{\pm }$. Quantity $B_{\pm }$ is singular on $r_{\gamma }^{\pm }$, and quantity $A_{\pm }$ is singular on $r_{mso}^{\pm }$ (where $A_{\pm }$ and $C_{\pm }$ are null on $r_{\gamma }^{\pm }$). However, they are regular on the inner edge in $r_{\mathrm{IN}}\in [r_{mbo}^{\pm },r_{mso}^{\pm }[$.

It is worth noting that for a disk (the limiting case of a RAD of the order $n=1$), considering a narrow ring with a Gaussian density distribution, the initial data for the density profile can be set as

$$\begin{array}{c}\hfill {\left.{\Sigma }_0\left(r\right)\right|}_{n=1}=e^{-{\displaystyle \frac{{(r-r_0)}^2}{b^2}}}\mathrm{with}{\left.\Sigma \left(r_{\mathrm{IN}}\right)\right|}_{n=1}=0\mathrm{and}{\left.\mathcal{F}\left(r_{\infty }\right)\right|}_{n=1}=0,\end{array}$$

(13)

where $r\in [r_{\mathrm{IN}},r_{\infty }]$ and for $r_{\mathrm{IN}}\le r_{mso}$ ($(r_0,b)$ are constant).

For a RAD, the initial density profile (${\Sigma }_0\equiv \Sigma (t=0,r)$) at $r\ge r_{\mathrm{IN}}$ is7

$$\begin{array}{c}\hfill {\Sigma }_d(0,r)\equiv \sum _{j=1}^n{\tilde{c}}_{j}exp\left[{\tilde{d}}_j{\left({\displaystyle \frac{r-{\tilde{r}}_j}{{\tilde{f}}_j}}\right)}^{{\tilde{p}}_j}\right],\end{array}$$

(14)

where $\{{\tilde{c}}_i,{\tilde{r}}_i,{\tilde{d}}_i,{\tilde{p}}_i\}$ are constants to be fixed for each model and can be written as follows:

$$\begin{array}{c}\hfill {\left.{\Sigma }_d(0,r)\right|}_{{\tilde{p}}_j=2,{\tilde{d}}_j=-1}\equiv \sum _{j=1}^n{q}_{j}exp\left[-{\left({\displaystyle \frac{r-r_{0_j}}{b_j}}\right)}^2\right].\end{array}$$

(15)

Here, $\{r_{0_j},b_j,q_j\}$ are constants, defining the RAD models $\left(\mathcal{Q}\right)$ listed in

Table 2. Initial tori models $\left(\mathcal{Q}\right)$ for the rings, defined in Equation (15). All the quantities are in dimensionless units. n is the RAD order. $r_{\mathrm{IN}}$ is the inner ring’s inner edge. $r_{0_j}$ is the central radius, $b_j$ is the initial disk spread and fixed $r_{0_j}$, and parameters $q_j$ and $b_j$ fix the disk mass. $\nu$ is the shear viscosity coefficient.

$\left(\mathbf{A}\right)$	$n=2$	$\{r_{0_1}=20,b_1=0.1,q_1=1/3\}$	$\{r_{0_2}=26,b_2=0.5,q_2=1/2\}$	$r_{\mathbf{IN}}=7$
$\left(\mathbf{B}\right)$	$n=1$	$\{r_{0_1}=36,b_1=2,q_1=1/2\}$	✓	$r_{\mathrm{IN}}=7$
$\left(\mathbf{C}\right)$	$n=2,\nu =2$	$\{r_{0_1}=10,b_1=0.1,q_1=1/3\}$	$\{r_{0_2}=24,b_2=1,q_2=1/2\}$	$r_{\mathrm{IN}}=2.21$
$\left(\mathbf{D}\right)$	$n=2,\nu =1$	$\{r_{0_1}=6,b_1=0.6,q_1=1/3\}$	$\{r_{0_2}=15,b_2=1.1,q_2=1/9\}$	$r_{\mathrm{IN}}=2.21$
$\left(\mathbf{E}\right)$	$n=2,\nu =1$	$\{r_{0_1}=32,b_1=0.71,q_1=1/3\}$	$\{r_{0_2}=26,b_2=1,q_2=1/3\}$	$r_{\mathrm{IN}}=8$
$\left(\mathbf{F}\right)$	$n=2,\nu =1$	$\{r_{0_1}=6,b_1=0.6,q_1=1/3\}$	$\{r_{0_2}=14,b_2=1.1,q_2=1/9\}$	$r_{\mathrm{IN}}=2.721$
$\left(\mathbf{G}\right)$	$n=2,\nu =2$	$\{r_{0_1}=10,b_1=0.1,q_1=1/3\}$	$\{r_{0_2}=24,b_2=1,q_2=1/2\}$	$r_{\mathrm{IN}}=2.721$
$\left(\mathbf{H}\right)$	$n=2,\nu =2$	$\{r_{0_1}=15,b_1=1,q_1=1/3\}$	$\{r_{0_2}=34,b_2=1,q_2=1/2\}$	$r_{\mathrm{IN}}=8$
$\left(\mathbf{I}\right)$	$n=2,\nu =2$	$\{r_{0_1}=26,b_1=0.1,q_1=1/3\}$	$\{r_{0_2}=36,b_2=0.5,q_2=1/2\}$	$r_{\mathrm{IN}}=6$
$\left(\mathbf{L}\right)$	$n=1,\nu =1$	$\{r_{0_1}=30,b_1=0.1,q_1=1/2\}$	✓	$r_{\mathrm{IN}}=6$
$\left(\mathbf{M}\right)$	$n=2,\nu =2$	$\{r_{0_1}=15,b_1=0.1,q_1=1/3\}$	$\{r_{0_2}=24,b_2=1,q_2=1/2\}$	$r_{\mathrm{IN}}=6$
$\left(\mathbf{N}\right)$	$n=2,\nu =1$	$\{r_{0_1}=39,b_1=0.51,q_1=1/3\}$	$\{r_{0_2}=18,b_2=0.1,q_2=1/9\}$	$r_{\mathrm{IN}}=6$
$\left(\mathbf{O}\right)$	$n=2,\nu =1$	$\{r_{0_1}=19,b_1=0.1,q_1=1/3\}$	$\{r_{0_2}=36,b_2=1,q_2=1/3\}$	$r_{\mathrm{IN}}=6$
$\left(\mathbf{P}\right)$	$n=2,\nu =1$	$\{r_{0_1}=24,b_1=0.6,q_1=1/3\}$	$\{r_{0_2}=12,b_2=1.1,q_2=1/9\}$	$r_{\mathrm{IN}}=6$
$\left(\mathbf{Q}\right)$	$n=2,\nu =2$	$\{r_{0_1}=18,b_1=0.1,q_1=1\}$	$\{r_{0_2}=28,b_2=0.1,q_2=1/2\}$	$r_{\mathrm{IN}}=6$
$\left(\mathbf{R}\right)$	$n=2,\nu =2$	$\{r_{0_1}=11,b_1=0.1,q_1=1/2\}$	$\{r_{0_2}=34,b_2=1,q_2=1/2\}$	$r_{\mathrm{IN}}=6$

).

In order to proceed with the numerical integration, we need to fix the spacetime and fluid parameters that regulate Equation (12). Therefore, in particular, we explored the spin range $a\in [0,1]$ by selecting two representative values of the spin, faster spinning and slower spinning BHs, integrating them numerically for all the cases combined, defined according to (1) different boundary conditions, (2) different initial density profiles, and (3) different viscosities. Hence, considering the discussion in Section 2.1, we explore RAD evolution in spacetimes with $a=0.2$ and $a=0.9$ (see Figure 1 and

Table 3. Details of the spacetime geodesic structure on the equatorial plane for Kerr BH spacetimes with spin $a=0.2$ and $a=0.9$, where the outer ergosurface is $r_{ϵ}^{+}=2$. Sign $(\pm )$ refers to counter-rotating/co-rotating orbits, respectively. $r_{mso}$ is the marginally stable orbit, $r_{mbo}$ is the marginally circular orbit, and $r_{\gamma }$ is the marginally circular orbit, which is also a photon circular orbit. Radii $r_{\left[\gamma \right]}$ and $r_{\left[mbo\right]}$ constrain the location of tori centers (points of maximum pressure in the disks) and are defined in Table 1. Radius $r_{+}$ is the outer BH horizon. All quantities are dimensionless.

$a=0.2$	$r_{\gamma }^{-}=2.76$	$r_{mbo}^{-}=3.59$	$r_{mso}^{-}=5.33$	$r_{\left[mbo\right]}^{-}=9.21$	$r_{\left[\gamma \right]}^{-}=18.48$
$r_{+}=1.98$	$r_{\gamma }^{+}=3.22$	$r_{mbo}^{+}=4.39$	$r_{mso}^{+}=6.64$	$r_{\left[mbo\right]}^{+}=11.67$	$r_{\left[\gamma \right]}^{+}=26.41$
$a=0.9$	$r_{\gamma }^{-}=1.558$	$r_{mbo}^{-}=1.732$	$r_{mso}^{-}=2.321$	$r_{\left[mbo\right]}^{-}=3.61$	$r_{\left[\gamma \right]}^{-}=4.887$
$r_{+}=1.436$	$r_{\gamma }^{+}=3.91$	$r_{mbo}^{+}=5.657$	$r_{mso}^{+}=8.717$	$r_{\left[mbo\right]}^{+}=15.59$	$r_{\left[\gamma \right]}^{+}=41.38$

). By fixing in this analysis two large and small values of BH spin, we evaluate (i) the effects of the different rotation orientation of each ring component on the orbiting ringed disk with respect to the central attractor, and (ii) the impact of the tori relative rotation in the disk evolution, which is emphasized for faster spinning attractors. By increasing the spin, differences are evident close to and far from the BH tori, according to the geodesic structure of the two spins. In this respect, we evaluate the case when the inner torus of a doublet is co-rotating versus counter-rotating (see Figure 1). Tori have different relative rotation orientations and different values of shear viscosity, with initial densities listed in Table 2. Given the problem of symmetries, we can study the co-rotating and counter-rotating rings considering $a=\pm 0.2$ and $a=\pm 0.9$, respectively, investigating ℓco-rotating seeds of co-rotating or counter-rotating rings, set by an initial (modulated) density profile ${\Sigma }_0$ according to Equation (15) for $n=2$ and within the models $\left(\mathcal{Q}\right)$ of Table 2. Then, we can investigate the ℓcounter-rotating rings, focusing on the co-evolution of each ℓco-rotating partial solution $\left(\mathcal{Q}\right)+\left(\mathcal{P}\right)$, where $\left(\right(\mathcal{Q}),(\mathcal{P}\left)\right)$ are two models of Table 2.

The following five sets of boundary conditions are explored8

$$\begin{array}{c}\left[\mathbf{1}\right]:\{\Sigma (t,r_{\mathrm{IN}})=0,{\partial }_r\Sigma (t,r_{\mathrm{IN}})=0,\mathcal{F}(t,r_{\infty })=0\};\left[\mathbf{2}\right]:\{{\partial }_r\Sigma (t,r_{\mathrm{IN}})=0,\mathcal{F}(t,r_{\infty })=0\};\hfill \\ \left[\mathbf{3}\right]:\{\Sigma (t,r_{\infty })=0,\mathcal{F}(t,r_{\infty })=0\};\left[\mathbf{4}\right]:\{\Sigma (t,r_{\infty })=0,\Sigma (t,r_{\mathrm{IN}})=0,{\partial }_r\Sigma (t,r_{\mathrm{IN}})=0\};\hfill \\ \left[\mathbf{5}\right]:\{\Sigma (t,r_{\mathrm{IN}})=0,\mathcal{F}(t,r_{\infty })=0\},\hfill \end{array}$$

(16)

where $\Sigma (0,t)={\Sigma }_0$, and the integration has been set in the range9$t\in [0,t_f]$ for $r\in [r_{\mathrm{IN}},r_{\infty }]$. For a single-ring model, we can evaluate the total disk mass as follows10

$$\begin{array}{c}\hfill {\displaystyle \frac{M_D}{2\pi }}={\int }_{r_{\mathrm{IN}}}^{r_{\infty }}r{U}^t\Sigma dr,\end{array}$$

(17)

and for n RAD components, we considered the contribution of all rings and the sum of each ring mass contribution, dividing into subsets the integration total range $[r_{\mathrm{IN}},r_{\infty }]$ (see Figure 22). The independent evolution of combined ℓcounter-rotating RAD seeds is studied for spin $a=\pm 0.9$ in Section 3.1 and Figures 3, 5, 7, 9 and 11), and for spin $a=\pm 0.2$, Section 3.2 and Figures 13, 15, 17, 19 and 24), focusing on the following composite systems:

–Figures 3, 9 and 11.

 The ℓco-rotating triplet of counter-rotating rings $(\mathbf{A}+\mathbf{B}):{\mathrm{C}}_{+}\left(\mathbf{A}\right)<{\mathrm{C}}_{+}\left(\mathbf{A}\right)<{\mathrm{C}}_{+}\left(\mathbf{B}\right)$.

The ℓcounter-rotating triplet $(\mathbf{B}+\mathbf{C}):{\mathrm{C}}_{-}\left(\mathbf{C}\right)<{\mathrm{C}}_{-}\left(\mathbf{C}\right)<{\mathrm{C}}_{+}\left(\mathbf{B}\right)$.

The ℓcounter-rotating triplet $(\mathbf{B}+\mathbf{D}):{\mathrm{C}}_{-}\left(\mathbf{D}\right)<{\mathrm{C}}_{-}\left(\mathbf{D}\right)<{\mathrm{C}}_{+}\left(\mathbf{B}\right)$.

The ℓco-rotating quadruplet of co-rotating rings $(\mathbf{D}+\mathbf{C}):{\mathrm{C}}_{-}\left(\mathbf{D}\right)<{\mathrm{C}}_{-}\left(\mathbf{C}\right)<{\mathrm{C}}_{-}\left(\mathbf{D}\right)<{\mathrm{C}}_{-}\left(\mathbf{C}\right)$.

–Figures 5 and 7.

 The ℓcounter-rotating quadruplet $(\mathbf{E}+\mathbf{F}):{\mathrm{C}}_{-}\left(\mathbf{F}\right)<{\mathrm{C}}_{-}\left(\mathbf{F}\right)<{\mathrm{C}}_{+}\left(\mathbf{E}\right)<{\mathrm{C}}_{+}\left(\mathbf{E}\right)$.

The ℓcounter-rotating quadruplet $(\mathbf{H}+\mathbf{G}):{\mathrm{C}}_{-}\left(\mathbf{G}\right)<{\mathrm{C}}_{+}\left(\mathbf{H}\right)<{\mathrm{C}}_{-}\left(\mathbf{G}\right)<{\mathrm{C}}_{+}\left(\mathbf{H}\right)$.

The ℓco-rotating quadruplet of co-rotating rings $(\mathbf{F}+\mathbf{G}):{\mathrm{C}}_{-}\left(\mathbf{F}\right)<{\mathrm{C}}_{-}\left(\mathbf{G}\right)<{\mathrm{C}}_{-}\left(\mathbf{F}\right)<{\mathrm{C}}_{-}\left(\mathbf{G}\right)$.

–Figures 13, 19, and 24.

 The ℓco-rotating triplet of counter-rotating rings $(\mathbf{I}+\mathbf{L}):{\mathrm{C}}_{+}\left(\mathbf{I}\right)<{\mathrm{C}}_{+}\left(\mathbf{L}\right)<{\mathrm{C}}_{+}\left(\mathbf{I}\right)$.

The ℓcounter-rotating triplet $(\mathbf{L}+\mathbf{M}):{\mathrm{C}}_{-}\left(\mathbf{M}\right)<{\mathrm{C}}_{-}\left(\mathbf{M}\right)<{\mathrm{C}}_{+}\left(\mathbf{L}\right)$.

The ℓco-rotating quadruplet of co-rotating rings $(\mathbf{M}+\mathbf{N}):{\mathrm{C}}_{-}\left(\mathbf{M}\right)<{\mathrm{C}}_{-}\left(\mathbf{N}\right)<{\mathrm{C}}_{-}\left(\mathbf{M}\right)<{\mathrm{C}}_{-}\left(\mathbf{N}\right)$.

The ℓcounter-rotating triplet $(\mathbf{L}+\mathbf{N}):{\mathrm{C}}_{-}\left(\mathbf{N}\right)<{\mathrm{C}}_{+}\left(\mathbf{L}\right)<{\mathrm{C}}_{-}\left(\mathbf{N}\right)$.

–Figures 15 and 17.

 The ℓcounter-rotating quadruplet $(\mathbf{O}+\mathbf{P}):{\mathrm{C}}_{-}\left(\mathbf{P}\right)<{\mathrm{C}}_{+}\left(\mathbf{Q}\right)<{\mathrm{C}}_{-}\left(\mathbf{P}\right)$ <${\mathrm{C}}_{+}\left(\mathbf{Q}\right)$.

The ℓco-rotating quadruplet of co-rotating rings $(\mathbf{Q}+\mathbf{P}):{\mathrm{C}}_{-}\left(\mathbf{P}\right)<{\mathrm{C}}_{-}\left(\mathbf{Q}\right)<{\mathrm{C}}_{-}\left(\mathbf{P}\right)<{\mathrm{C}}_{-}\left(\mathbf{Q}\right)$.

The ℓcounter-rotating quadruplet $(\mathbf{R}+\mathbf{Q}):{\mathrm{C}}_{+}\left(\mathbf{R}\right)<{\mathrm{C}}_{-}\left(\mathbf{Q}\right)<{\mathrm{C}}_{-}\left(\mathbf{Q}\right)<{\mathrm{C}}_{+}\left(\mathbf{R}\right)$.

As is also confirmed in the analysis of [17], the diffusive evolution of a ringed disk can be described in four phases (analogue to a RAD evolution in a Keplerian frame). Initially, we observe the rings spreading inside the RAD. Consequently, the internal ringed structure changes by leveling the density differences among the disks. This phase is characterized by marked internal activity with interaction among each ringed component. Following this evolution, an internal density maximum appears, signaling the formation of a single disk. The final phase is, therefore, dominated by single-disk structure evolution. The predominant dynamics at the origin are regulated mainly by the RAD boundary conditions.

Concentrating on the early stages of RAD evolution, we will focus on the description of the phases preceding the disappearance of the internal structure, distinguishing the RAD internal dynamics with their inter-disk activity. In this phase, the densities maximum generally spreads towards the origin, and the disk’s outer edges move to larger radii. Note, assuming the inner edge is close to the marginally stable orbit, a ring evolution (at small $\Sigma$) proceeds similarly to the non-relativistic setup. On the other hand, the low viscosity approximation cannot hold well for the later stages of ring cluster evolution. It is clear that the accreting matter from the inner edge of the outer ring of the cluster can have an unperturbed radial component ${\tilde{U}}^r\ne 0$11.

In Section 3.1, we consider RADs orbiting a central attractor with spin $a=\pm 0.9$, and in Section 3.2, RADs orbiting a slower spinning BH with spin $a=\pm 0.2$ are explored.

3.1. BH Attractors with Spin $a=\pm 0.9$

In this section, we consider RAD orbiting BH attractors with spin $a=\pm 0.9$ (see Table 3). The ℓco-rotating rings surface density evolution in the BH spacetimes with $a=\pm 0.9$ is shown in Figure 2; co-rotating and counter-rotating fluids are also discussed. In this case, the models $\left\{\right(\mathbf{A}),(\mathbf{B}),(\mathbf{C}),(\mathbf{D}\left)\right\}$ of Table 2 are considered (note, model $\left(\mathbf{B}\right)$ (upper-right panel) is composed of one counter-rotating ring), with the boundary condition $\left[\mathbf{1}\right]$ of Equation (16) and different values of the viscosity coefficient $\nu$. With the analysis of the model $\left(\mathbf{B}\right)$, we can compare single-ring diffusive evolution with RAD diffusive evolution. In condition $\left[\mathbf{1}\right]$, seed evolution is characterized by a building-up of matter close to the origin (inner edge of the inner RAD ring) at later stages of evolution (more evident in the counter-rotating case). Focusing on the initial phases of seed evolution featuring RAD inner structure evolution with ring interaction, the process timescales of the inner ringed structure re-modulation depend on initial ring spreading and density (the process appears less dependent on different viscosity values). The first phases of spread and merging are fast, rapidly followed by the formation of one maximum density for a newly formed one-ring structure located close to the initial density maximum with a radial drift velocity. The disk in this phase can retain a modulated inner density profile depending on the tori’s initial spreading and separation (see model $\left(\mathbf{D}\right)$). In the analysis of combined seeds in Figure 3, we also consider the combined ℓco-rotating rings quadruplets or triplets compared with the case of the ℓco-rotating solutions of RAD diffusive equations.

The combination of the ℓcounter-rotating seeds in Figure 3 shows that the evolution is qualitatively similar to the ℓco-rotating seed evolution. In this combined analysis, the radial range of integration is fixed considering the two seeds’ different inner edges. Interestingly, there is the permanence of the modulated inner structure at later times, with the inner dynamics not affecting the inner (and outer) RAD region’s evolution.

The boundary condition $\left[\mathbf{2}\right]$ of Equation (16) has been tested in Figure 4. In the $\left(\mathbf{F}\right)$ model, we note a phase of matter decreasing at the seed’s inner edge, which can also be seen in the seed (ℓcounter-rotating) combination in Figure 5.

In Figure 6, conditions $\left[\mathbf{3}\right]$ of Equation (16) are tested. The building-up phase at the inner edge is evidenced together with the (rapid) destruction of the RAD inner structure, also characterizing the ℓcounter-rotating RAD in the combined evolution of Figure 7. The persistence of a modulated inner-density structure at the stage of one single disk could be seen as fingerprint of a RAD initial stage, constituted by ring aggregates characterized by different viscosity prescriptions.

Boundary conditions $\left[\mathbf{4}\right]$ and $\left[\mathbf{5}\right]$ of Equation (16) are tested in Figure 8 and Figure 9, respectively, for the initial density profiles $\left\{\right(\mathbf{A}),(\mathbf{B}),(\mathbf{C}),(\mathbf{D}\left)\right\}$ of Table 2. Remarkably, the RAD seed evolution is then similar to the evolution in Figure 2.

Different (inner) boundary conditions on the RAD edges affect mostly the RAD evolution rather then the inner structure evolution generally resolved in the early times (see, for example, Figure 10 and Figure 11).

3.2. BH Attractors with Spin $a=\pm 0.2$

In Figure 12 and Figure 13, we consider clusters orbiting BHs with spin $a=\pm 0.2$. Similarly to the case in Figure 2, boundary conditions $\left[\mathbf{1}\right]$ of Equation (16) are considered here for the initial density profiles $\left\{\right(\mathbf{I}),(\mathbf{L}),(\mathbf{M}),(\mathbf{N}\left)\right\}$ of Table 2, where model $\left(\mathbf{L}\right)$ (upper-right panel) is composed of one counter-rotating ring. The destruction of the inner RAD structure is followed by the formation of an inner maximum density also present during later stages of RAD evolution.

Boundary conditions $\left[\mathbf{2}\right]$ are tested in Figure 14 and Figure 15, where, for the $\left(\mathbf{P}\right)$ model, an increase in mass at the inner edge followed by a decreasing phase during later stages of evolution is noted.

From the analysis of the boundary condition $\left[\mathbf{3}\right]$ applied to the $\left\{\right(\mathbf{O}),(\mathbf{P}),(\mathbf{Q}),(\mathbf{R}\left)\right\}$ models of Figure 16, we see that, after the leveling of the inner ringed structure, the single maximum density of the final disk appears to be affected mostly by the spreading of the initial ring cluster.

Figure 17 shows the independent evolution of ℓco-rotating rings with different viscosities. RAD evolution does not differ qualitatively from the ℓco-rotating solutions of the diffusive RAD equation.

In Figure 18 and Figure 19, boundary conditions $\left[\mathbf{4}\right]$ of Equation (16) are applied to the models $\left\{\right(\mathbf{I}),(\mathbf{L}),(\mathbf{M}),(\mathbf{N}\left)\right\}$ and can be compared to Figure 8.

In Figure 20, $\mathcal{F}$ (the mass flux defined in Equation (11) in the Kerr spacetime with $a=\pm 0.2$ for the ℓco-rotating rings is shown at different stages of RAD evolution. Boundary condition $\left[\mathbf{4}\right]$ of Equation (16) has been adopted with initial models $\left\{\right(\mathbf{I}),(\mathbf{L}),(\mathbf{M}\left)\right\}$ of Table 2. The related density evolution is shown in Figure 18). The flux zeros (changing slowly with time) as well as the disk spreading during RAD evolution and the destruction of the inner ringed density profile are evidenced.

In Figure 21, the radial drift of the fluid density $\Sigma$ is considered by studying the solution ${\Sigma }^{\prime }\equiv {\partial }_r\Sigma =0$ in the plane $t-r$ for co-rotating and counter-rotating fluids orbiting the BH spacetime with spin $a=\pm 0.2$, respectively. The initial density profiles are models $\left\{\right(\mathbf{I}),(\mathbf{L}),(\mathbf{M}\left)\right\}$ (density evolution is shown in Figure 18), with the boundary condition $\left[\mathbf{4}\right]$. The integration ranges consider the RAD inner $\left(\mathbf{i}\right)$, center $\left(\mathbf{c}\right)$, and outer $\left(\mathbf{o}\right)$ tori. The systems $\left(\mathcal{Q}\right)+\left(\mathcal{P}\right)$ for the two general models $\left\{\right(\mathcal{Q}),(\mathcal{P}\left)\right\}$ consider sums that are two seed evolutions apart, which is the solution of ${\Sigma }^{\prime }\left(\mathcal{Q}\right)=0$ and ${\Sigma }^{\prime }\left(\mathcal{P}\right)=0$, while system $(\mathcal{Q}+\mathcal{P})$ is the solution of ${\Sigma }^{\prime }\left(\mathcal{Q}\right)+{\Sigma }^{\prime }\left(\mathcal{P}\right)=0$. The radial ranges distinguishing the $\left(\mathbf{i}\right)$, $\left(\mathbf{c}\right)$, and $\left(\mathbf{o}\right)$ components are defined by the radii $r_{\mathrm{IN}}<r_{\left(c\right)}\le r_{*}<r_{\infty }$. The radial drift of the inner and outer tori follows the destruction of the inner ringed structure12

In Figure 22, the evolution of the disk mass $M_D$ of Equation (17) is studied as a function of the dimensionless time. Co-rotating and counter-rotating fluids were considered orbiting the BH spacetime with spin $a=\pm 0.2$, respectively (see Table 3). The initial density profiles are the models $\left\{\right(\mathbf{I}),(\mathbf{L}),(\mathbf{M}\left)\right\}$ of Table 2, with the boundary condition $\left[\mathbf{4}\right]$ of Equation (16), and the related density evolution is shown in Figure 18. The systems $\left(\mathcal{Q}\right)+\left(\mathcal{P}\right)$ for two general models $\left\{\right(\mathcal{Q}),(\mathcal{P}\left)\right\}$ represent the sums of the two evolutions apart, which is the sum of $M_D\left(\mathcal{Q}\right)/2\pi$ and $M_D\left(\mathcal{P}\right)/2\pi$, while $(\mathcal{Q}+\mathcal{P})$ is the mass of the composed system formed by $\left(\mathcal{Q}\right)$ and $\left(\mathcal{P}\right)$, where these two evaluations are clearly coincident. At later stages of the RAD disk evolution, the curves decrease approximately with a power law ($t^s$ for $s =$ constant). The integration ranges consider the RAD inner $\left(\mathbf{i}\right)$, center $\left(\mathbf{c}\right)$, and outer $\left(\mathbf{o}\right)$ tori. There is, therefore, an exponential decrease in RAD mass at a later stage of RAD evolution, similar to one disk evolution. The decomposition with the selection of the radii $r_c<r_{*}$ is set here on the condition of faint density in the ring’s interspace. The analysis of the RAD decomposition for the ℓco-rotating couples or the combined evolution of the RAD seeds in the inner, outer, and eventually, central ring shows the mass increase on one ring component in correspondence to the mass decreases of the other inner component following matter exchange for accretion from the outer ring to the inner one, for example, from the outer disk to the central ring or the outer ring to the inner ring, as shown in Figure 18. The outer ring mass increases at larger stages of evolution independently to the disk rotation orientation with respect to the Kerr BH and the relative rotation orientation.

The analysis of the RAD seed diffusive equation in the R-frame is completed in Figure 23 and Figure 24 with the initial density profiles $\left\{\right(\mathbf{I}),(\mathbf{L}),(\mathbf{M}),(\mathbf{N}\left)\right\}$ of Table 2 with the boundary condition $\left[\mathbf{5}\right]$ of Equation (16). It is interesting to note that the persistence of the maximum density at later stages of evolution depends on the ring’s spreading rather then the ring’s density maximum.

4. Conclusions

We discussed the evolution equation of a ringed accretion disk composed of clusters of co-rotating and counter-rotating general relativistic thin disks, solving a diffusion-like propagating equation for the RAD surface density. Following [1], the approach developed for a single disk is applied here to a ring agglomerate orbiting on the equatorial plane of a central Kerr BH and defined by an initial modulated density profile. The clusters are composed of rings of different densities and spreading. The numerical solutions of the RAD diffusive equations, recovered from the conservation of particle number and the toroidal component of the stress-energy tensor, have been found to test multiple viscosity prescriptions and boundary conditions. The analysis was performed considering both faster-spinning and slower-spinning BH spacetimes. The initial spreading of disks, constituting the internal ringed structure, governs the RAD evolution, being a predominant factor with respect to the initial ring densities.

It was found that the first stage of the cluster evolution involves inner structure dynamics and is characterized by spreading in the RAD and the modification of the internal structure disappearing in the second stage, while the third stage is dominated by the formation of a single disk with an internal density maximum. The final stage of RAD evolution is characterized by single-disk evolution at the origin. We focused mostly on the first stages of evolution. Close to the last stable circular radius, for small density $\Sigma$, our analysis shows that the disks evolve similarly to the non-relativistic case (see also [17]).

The investigation first focused on the evolution of a set of ℓcounter-rotating or ℓco-rotating sets subjected to several viscosity prescriptions and boundary conditions, defined by an initial modulated density profile governing the inner ringed structure of the RAD. We then analyzed the combined independent evolution of the ℓcounter-rotating seeds. In this combined analysis, the radial range of integration was fixed considering the two seeds’ different inner edges. The combination of the ℓcounter-rotating seeds shows that the evolution is qualitatively similar to the ℓco-rotating evolution. However, we proved the persistence of an inner ringed structure also at later times, while the internal dynamics (characterized by ring component interaction) do not affect the RAD boundary region’s evolution.

Hence, the analysis explores wide ranges of different cases, selected in the classes (of possible configurations) relevant to the phenomenological impact, especially for the astrophysical context of SMBHs and their accretion disk evolution. Therefore, we analyzed, among the different cases, the following: (1) differences for fast and slowly spinning BHs, (2) the impact of tori number, (3) the divergences between ℓco-rotating and ℓcounter-rotating tori, and (4) the effects of different viscosities and boundary conditions. The astrophysical differences evidenced in the disk density evolution for each different case are dominated by the different relative rotations of the tori and the density differences in the initial ringed density. These aspects affect the timescales of the process and characterize the final phases of RAD evolution, which are constituted by one single disk resulting from the multi-tori merger. We go into detail below, summarizing the impact of each different condition on the final evolution. Note that the aggregates of different disks with different viscosity are equivalent, in our frame, to a single disk affected by inhomogeneous viscosity (depending on the radius) and with possibly different rotation orientation in the (single) ringed profile. However, for the astrophysical context, the most impacting feature appears during composed (independent) evolution, which can be split into the contribution of different components (as in Figure 22), evident in the mass accretion rate. The effects of disk morphology as the set of several systems are then evident in all the phenomena associated with the existence of the multiple inter-disk inner-edges. The relative rotation and initial difference in density, differentiated particularly in the spacetimes of the faster spinning attractors, are, also in this frame, the most relevant factors influencing the process timescales.

The evolution of the disk mass $M_D$, mass flux, and radial drift was also studied as a function of the dimensionless time, in dependence on the ring rotation orientation, BH spin, and the initial RAD density profile. The mass increases on one ring component, and in correspondence, the mass decreases the other inner component, mostly independent from the ring’s relative rotation orientation and rotation with respect to the central BH. At later stages, the curves $M_D$ decrease approximately with a power law ($t^s$ for $s=$ constant) (within the condition of faint density in the rings interspace). On the other hand, focusing on the initial stages of the seed evolution featuring the RAD inner structure evolution with rings interaction, the process timescales of the inner ringed structure re-modulation depend on the initial rings spreading and density (the process appears less dependent on the different viscosity values, in the first approximation the viscosity appears most relevant factor when the viscosity fluctuation in the RAD distribution is the greater). The disk in this stage can retain a modulated inner density profile depending on the tori’s initial spreading and separation (see, for example, model $\left(\mathbf{D}\right)$).

However, it should be noted that, while in this analysis stress models have been applied in the entire range of integration, other conditions should apply at infinity, particularly in the inner region (for $r\in ]r_{+},r_{\mathrm{IN}}[$). In fact, in the RAD, the perturbation analysis is complicated by the typical inter-disk activity constituted by the inter-ring accretion and disk collision. More generally, the presence of multiple inner edges, due to the internal ringed structure, undermines the validity of the use of viscosity as a perturbation for the only non-zero radial fluid velocity. That is, it exists as an unperturbed component ${\tilde{U}}^r\ne 0$ in the presence of radial inter-disk flux.

Furthermore, for fast-spinning attractors ($a>0.828427$), the co-rotating ring is expected to partially orbit the outer ergoregion of the Kerr spacetime, and for faster-spinning attractors ($a>0.942809$), the inner edge of the inner co-rotating ring is expected to be in the outer-ergoregion, and the ring can be totally contained in this region. In this situation, the counter-rotating flow from the outer disk, having an initial toroidal velocity $u^{\varphi }<0$, arrives at the outer ergosurface with $u^{\varphi }>0$. The flow is free-falling and there is a constant $\ell <0$ along all the motion. Therefore, the flow crosses, and before reaching the outer ergosurface, an inversion surface embeds the central BH, where $u^{\varphi }=0$, which is defined by the parameter ℓ. Consequently, in the presence of an inner co-rotating disk, the counter-rotating flow from the outer torus can impact the inner ring with $u^{\varphi }>0$ [21].

However, within all these approximations, our analysis is a feasible test of the early stages of RAD evolution. For all these reasons, this early exploratory study provided a preliminary but well-founded overview of the multi-ring frame of ring aggregate evolution.

We stress the possibility of accretion rings being produced by tidal disruptions of multiple stellar systems when they sink too close to the SMBH. In this frame, our application could be extremely timely in connection with the recent apparent detection of the the first binary star in the S-cluster close to SgrA* [22].