Ringed Accretion Disks: Evolution of Double Toroidal Configurations

The physics of accretion disks around supermassive attractors is characterized by several processes of very diverse natures, and it is the foundation of many phenomena in high-energy astrophysics. However, the existence of a unique satisfactory framework for a complete theoretical interpretation of such observations still remains to be proven, as the phenomenology associated with these systems includes very different events all supposedly related to the physics of strong attractors and their environment: from the issue of jet generation and collimation, to gamma-ray bursts (GRBs) and the accretion process itself. In particular, the accretion of matter from disks orbiting supermassive black holes (SMBHs), hosted in the center of active galactic nuclei (AGNs) or quasars, is a subject that seems to require consistent additional investigations. The location and definition of the inner edge of an accretion disk, for example, are topics that have been continuously debated, with several theories often leading to different conclusions (Bromley et al. 1998; Agol & Krolik 2000; Paczyński 2000; Krolik & Hawley 2002; Abramowicz et al. 2010). The interaction between SMBHs and orbiting matter, which entangles the central attractor and the embedded material in one dynamical picture, is certainly a complicated subject of investigation. The interaction between the attractor and the environment could potentially give rise to a mutation of the geometrical characteristics of spacetime, which, initially considered as "frozen background", may change following a spin-down or eventually spin-up of the non-isolated black hole (BH), could potentially give rise to a mutation of the geometrical characteristics of spacetime (Abramowicz et al. 1983, 1998; Font & Daigne 2002a; Rezzolla et al. 2003; Hamersky & Karas 2013; Korobkin et al. 2013). Jet emission is then a further key element of such systems: the relation between the rotational energy of the central attractor, the disk's inner edge location, and jet emission (jet–accretion correlation) is still substantially obscure (Abramowicz & Sharp 1983; Marscher et al. 2002; Maraschi & Tavecchio 2003; Ferreira & Casse 2004; Okuda et al. 2005; Allen et al. 2006; Lyutikov 2009; Maitra et al. 2009; Fragile et al. 2012; McKinney et al. 2013; Coughlin & Begelman 2014; Ghisellini et al. 2014; Lovelace et al. 2014; Sbarrato et al. 2014; Chen et al. 2015; Fender & Munoz-Darias 2015; Sadowski & Narayan 2015; Yu et al. 2015; Zhang et al. 2015; Stuchlík & Kolos 2016).

However, considering the huge variety of approaches to and studies on each of these individual issues, it appears necessary, in addressing these topics, to formulate the investigation in a more global perspective. By reframing the problem in terms of structures and macro-structures, the first understood as isolated objects and the second as isolated clusters of individual interacting objects, the relation between the different components in these systems can be considered, and the knowledge acquired on certain specific processes, such as the accretion mechanism, jet properties, and black hole physics, can be included in a more comprehensive picture.

Accretion disk models may be differentiated based on at least three important aspects: the geometry (the vertical thickness defines geometrically thin or thick disks), the matter accretion rate (sub- or super-Eddington luminosity), and the optical depth (transparent or opaque disks)—see Abramowicz & Fragile (2013). Geometrically thick disks are well-modeled by Polish Doughnuts (P-Ds) (Abramowicz et al. 1978; Kozlowski et al. 1978; Jaroszynski et al. 1980; Stuchlík et al. 2000; Rezzolla et al. 2003; Slaný & Stuchlík 2005; Stuchlik 2005; Pugliese et al. 2012; Pugliese & Montani 2013 or by ion tori (Rees et al. 1982). P-D tori (with very high, super-Eddington, accretion rates) and slim disks have high optical depth, while Ion Tori and ADAF (advection-dominated accretion flow) disks have low optical depth and relatively low accretion rates (sub-Eddington). Geometrically thin disks are generally modeled by the standard Shakura–Sunyaev (Keplerian) disks (Novikov & Thorne 1973; Page Don & Thorne Kip 1974), ADAF disks (Narayan et al. 1998; Abramowicz & Straub 2014), and slim disks (Abramowicz & Fragile 2013). In these disks, dissipative viscosity processes are relevant for accretion, and are usually attributed to the magnetorotational instability of local magnetic fields (Hawley et al. 1984; Hawley 1987, 1991; De Villiers & Hawley 2002). On the other hand, in toroidal disks, pressure gradients are crucial (Abramowicz et al. 1978). As proposed by Paczyński (1980), accretion disks can be modeled by using an appropriately defined pseudo-Newtonian potential (see also Novikov & Thorne 1973; Abramowicz et al. 1978; Stuchlik 2005; Stuchlik et al. 2009; Abramowicz & Fragile 2013).

In Pugliese & Stuchlík (2015), we considered the possibility that over several accretion regimes that occurred in the lifetime of a non-isolated supermassive Kerr black hole, several toroidal fluid configurations might have been formed from the interaction of the central attractor with the environment in AGNs, where corotating and counterrotating accretion stages are mixed (Lovelace & Chou 1996; Alig et al. 2013; Carmona-Loaiza et al. 2015; Dyda et al. 2015; Gafton et al. 2015). These systems can then be re-animated in subsequent stages of the BH's accretion disk's life, for example, in colliding galaxies, in the galactic center, in some kinds of binary systems where additional matter could be supplied in the vicinity of the central black hole by the tidal distortion of a star, or if some cloud of interstellar matter is captured by the strong gravity.

We formulated an analytic model of a macro-structure, the ringed accretion disk, composed of several toroidal axisymmetric subconfigurations (rings) of corotating and counterrotating fluid structures (tori) orbiting one central supermassive Kerr BH, with the symmetry plane coinciding with the equatorial plane of the central Kerr BH. The emergence of instabilities for each ring and the entire macro-structure was then addressed in Pugliese & Stuchlík (2016) and D. Pugliese & Z. Stuchlík (2016, in preparation). Similar studies on analogue problems are in Cremaschini et al. (2013), where off-equatorial tori around compact objects were considered, and also in Nixon et al. (2012b). In D. Pugliese & Z. Stuchlik (2017, in preparation), we drew some conclusions for the case of only two toroidal disks orbiting a central Kerr attractor. We demonstrated that a double accretion system may only be formed under specific conditions. The rings of the macro-structure can then interact by colliding. The center-of-mass energy during ring collision was evaluated within the test particle approximation, demonstrating that the energy efficiency of the collisions increases with increasing dimensionless BH spin, which is very high for near-extreme BHs. The collisional energy efficiency could be even higher in near-extreme Kerr naked singularity spacetimes (Stuchlík & Schee 2013, 2012; Stuchlik 1980).

Using numerical methods, multi-disks have also been analyzed in more complex, non-symmetric situations. The formation of several accretion disks in the geometries of the SMBH in AGNs or in binary systems has been considered in relation to various factors, where the rupture of symmetries has been addressed, for example, for titled, warped, non-coplanar disks. Attention has been paid to the investigations of the relevance of the disk geometry in attractor–disk interactions. The initial stages of the formation of such systems have been addressed in Ansorg et al. (2003). Regarding counteraligned accretion disks in AGNs, we point out the work of King & Pringle (2006), where the Bardeen–Petterson effect is proposed as a possible cause of the counteralignment of BH and disk spins: it was shown that BHs can grow rapidly if they acquire most of the accreted mass in a sequence of randomly oriented accretion episodes. In Lodato & Pringle (2006), the evolution of misaligned accretion disks and spinning BHs are considered specifically in AGNs, where the BH spin changes under the action of disk torques, as the disk, being subjected to Lense–Thirring precession, becomes twisted and warped. It was shown that accretion from misaligned disks in galactic nuclei would be significantly more luminous than accretion from flat disks. The aligning of Kerr BHs and accretion disks are studied in King et al. (2005). In Nealon et al. (2015), the effects of BH spin on warped or misaligned accretion disks are studied in connection to the role of the inner edge of the disk in the alignment of the angular momentum with the BH spin. The stable counteralignment of a circumbinary disk is the focus in Nixon (2012). King et al. (2008) argue that there is a generic tendency of AGN accretion disks to become self-gravitating at a certain radius from the attractor. The study of particular accretion processes, including the merging of the AGN accretion disk, demonstrates that the disk generally has a lower angular momentum than the BH; for an analogous limit, see Pugliese & Montani (2015). The chaotic accretion in AGNs could produce counterrotating accretion disks or strongly misaligned disks with respect to the central SMBH spin. Rapid AGN accretion from counterrotating disks is addressed in Nixon et al. (2012a) in particular. The authors studied the angular momentum cancellation in accretion disks characterized by a significant tilt between the inner and outer disk parts. These studies show that the evolution of misaligned disks around a Kerr BH might lead to the tearing up of the disk into several planes with different inclinations. The tearing up of a disk in misaligned accretion into a binary system is considered, for example, in Nixon et al. (2013).

The tearing-up process has also been considered as a possible mechanism behind the almost-periodic emission in the X-ray emission band known as quasi-periodic oscillations (QPOs). Tearing up a misaligned accretion disk with a binary companion is addressed in Dogan et al. (2015). Disk formation by tidal disruptions of stars on eccentric orbits by a spherically symmetric BH is considered in Bonnerot et al. (2016). For misaligned gas disks around eccentric BH binaries, see Aly et al. (2015).

As explained in Nixon et al. (2012b), in realistic cases of AGN accretion or also in stellar-mass X-ray binaries, there is a break in the central part of the tilted accretion disks orbiting Kerr BHs, due to the Lense–Thirring effect. The disk is thus split into several, essentially separated, planes. It is also observed that for small tilt angles, the disk may still break, and this must be connected with some observable phenomena as, for example, QPOs. For a brief review of the SMBH accretion mergers and accretion flows onto SMBHs, see King & Nixon (2013).

The existence of ringed disks in general may lead us to reinterpret phenomena so far analyzed in a single disk framework in terms of orbiting multi-toroidal structures. Specifically, this shift could be reinforced in modeling the spectral features of multi-disk structures. It is generally assumed that the X-ray emission from AGNs is related to accretion disks and the surrounding coronae. Assuming this is related to accretion disk instabilities, the spectral interpretation of X-ray emission is taken to constrain the main BH disk model parameters. We argue that this spectral profile should also provide a fingerprint of the ringed disk structure, possibly showing as a radially stratified emission profile. In fact, the simplest structures of this kind are thin radiating rings. The signature of alternative gravity, as exotic objects, given by spectral lines from the radiating rings is investigated in Schee & Stuchlik (2009), Schee & Stuchlik (2013), Bambi et al. (2016), and Ni et al. (2016). Sochora et al. (2011) propose that models of BH accretion rings may be revealed by future X-ray spectroscopy from the study of relatively indistinct excesses on top of relativistically broadened spectral line profiles, which, unlike the main body of the broad line of the spectral line profile, are connected to an extended (continuous) region of the accretion disk. They predicted relatively indistinct excesses of the relativistically broadened emission-line components arising in a well-confined radial distance from the accretion disk, thereby envisaging a sort of ring model that may be adapted as a special case of the model discussed in Pugliese & Stuchlík (2015, 2016a). In Karas & Sochora (2010), extremal energy shifts of radiation from a ring near a rotating BH were studied: radiation from a narrow circular ring shows a double-horn profile with photons having energy around the maximum or minimum of the range (see also Schee & Stuchlik 2009). This energy span of spectral lines is a function of the observer's viewing angle, the BH spin, and the ring radius. The authors describe a method to calculate the extremal energy shift in the regime of strong gravity. The accretion disk is modeled by rings located in a Kerr BH equatorial plane, originating from a series of episodic accretion events. It is argued that the proposed geometric and emission ringed structure should be evident from the extremal energy shifts of each ring. Accordingly, the ringed disks may be revealed through detailed spectroscopy of the spectral line wings. Although the method has been specifically adapted for the case of geometrically thin disks, an extension to thick rings should be possible. Furthermore, as detailed in Pugliese & Stuchlík (2016a, 2015), some of the general geometric characteristics of the ringed disk structure are applicable to the thin disk case.

Here we extend the study in Pugliese & Stuchlík (2015, 2016a) by considering an orbiting pair of axisymmetric tori governed by the relativistic hydrodynamic Boyer condition of equilibrium configurations of rotating perfect fluids (Boyer 1965). Our primary result is the characterization of the rings–attractor systems in terms of equilibrium or unstable (critical) topology, constraining the formation of such a system on the basis of the (frozen) dimensionless spin–mass ratio of the attractor and the relative rotation of the fluids. We investigate the possible dynamical evolution of the tori, generally considered as the transition from the topological state of equilibrium to a topology of instability, and the evolution for the entire macro-configuration, when accretion onto the central BH and collision among the tori may occur. We illustrate the scenario where tori collisions lead to the destruction of the macro-configuration. We summarized this analysis by developing evolutionary schemes that provide indications of the topology transition and the situations where these systems could potentially be found and then observed due to the associated phenomena. These schemes are constrained by the spin of the attractor and the relative rotation of the rings with respect to the attractor or each other. From a methodological viewpoint, we represented the evolutionary schemes with graph models, which we also consider here as references for our discussion.

In our model, we primarily evaluate the general relativistic effects on the orbiting matter in those situations where there are considerable curvature effects and fluid rotation which determine the toroidal topology and morphology. We focus on the  toroidal disk model orbiting supermassive Kerr attractors using the geometrically thick disk P-D, opaque and with very high (super-Eddington) accretion rates, where pressure gradients are crucial (Abramowicz et al. 1978; Kozlowski et al. 1978; Jaroszynski et al. 1980; Stuchlík et al. 2000; Rezzolla et al. 2003; Slaný & Stuchlík 2005; Stuchlik 2005; Pugliese et al. 2012). These configurations are often adopted as the initial conditions in the set-up for simulations of MHD (magnetohydrodynamic) accretion structures (Igumenshchev & Abramowicz 2000; De Villiers & Hawley 2002; Sragile et al. 2007). In fact, the majority of current analytical and numerical models of accretion configurations assumes axial symmetry of the extended accreting matter.

For geometrically thick configurations, it is generally assumed that the timescale of dynamical processes, ${\tau }_{\mathrm{dyn}}$ (regulated by the gravitational and inertial forces; the timescale for pressure to balance the gravitational and centrifugal forces), is much lower than the timescale of thermal ones, ${\tau }_{\mathrm{the}}$ (i.e., heating and cooling processes; the timescale of radiation entropy redistribution), which is lower than the timescale of viscous processes, ${\tau }_{\mathrm{vis}}$ (the effects of strong gravitational fields are dominant with respect to dissipative ones and predominantly determine the unstable phases of the systems; Igumenshchev & Abramowicz 2000; Font & Daigne 2002b; Abramowicz & Fragile 2013), i.e., ${\tau }_{\mathrm{dyn}}\ll {\tau }_{\mathrm{the}}\ll {\tau }_{\mathrm{vis}}$; see also Sragile et al. (2007), De Villiers & Hawley (2002), Hawley (1987, 1991), and Hawley et al. (1984). This in turn is the basis of the assumption of the perfect fluid energy–momentum tensor. Thus, the effects of strong gravitational fields dominate those of the dissipative ones (Paczyński 1980; Font & Daigne 2002b; Abramowicz & Fragile 2013). Consequently, during the evolution of dynamical processes, the functional form of the angular momentum and entropy distribution depends on the initial conditions of the system and on the details of the dissipative processes. Paczyński realized that it is physically reasonable to assume an ad hoc distribution (Abramowicz 2008). This feature constitutes a great advantage of these models and renders their adoption extremely useful and predictive (angular momentum transport in fluids is perhaps one of the most controversial aspects of thin accretion disks). Moreover, we should note that the Paczyński accretion mechanics from Roche lobe overflow induces the mass loss from tori, which is an important local stabilizing mechanism against thermal and viscous instabilities, and globally against the Papaloizou–Pringle instability (for a review, we refer the reader to Abramowicz & Fragile 2013).

In these models, the entropy is constant along the flow. According to the von Zeipel condition, the surfaces of constant angular velocity Ω and of constant specific angular momentum ℓcoincide (Abramowicz 1971; Chakrabarti 1990, 1991; Zanotti & Pugliese 2015), and the rotation law ${\ell }={\ell }({\rm{\Omega }})$ is independent of the equation of state (Abramowicz 2008; Lei et al. 2008).

Article layout. The plan of this article is as follows. The introduction of the thick accretion disk model in Kerr spacetime is summarized in Section 2, where the main notation used throughout this work is presented. This section constitutes first the introductory part of this work and also the disclosure of the methodological tools used throughout. We provide the main definitions of the major morphological features of the ringed disks. Then, we specialize the concepts for the case of a system composed of only two tori. After writing the Euler equations for the orbiting fluids, we cast the set of hydrodynamic equations for the tori by introducing an effective potential function for the macro-configuration. We then investigate the parameter space for this model: one set of coupled parameters provides the boundary conditions for the description of two tori in the macro-configuration. We proceed by dividing the discussion into ℓcorotating and ℓcounterrotating tori—if the tori are both corotating or counterrotating with respect to the central Kerr BH, they are ℓcorotating; if one torus is corotating and the other counterrotating they are ℓcounterrotating. However, even in the case of one pair of tori orbiting around a single central Kerr BH, a remarkably large number of possible configurations is possible. Therefore, in order to simplify and illustrate the discussion, we made use of special graphs to represent a couple of accretion tori and their evolution, within the constraints to which they are subjected. The use of these graphic schemes has been reveled to be crucial for the study and representation of these evolutionary cases. Although the following analysis may be followed quite independently from the graph formalism, they can also be used to quickly collect the different constraints on the existence and evolution of the tori and for reference in our discussion. Therefore, we also include here a brief description of the graph construction and basic concepts related to these structures. Appendix A discloses the details on the construction and interpretation of the graphs. The main graph blocks are listed in Figure 6. The main analysis of the present work is in Section 3, where the double-torus disk system is discussed in detail. We particularize the investigation detailing the double system on the basis of the relative rotation of fluids in the disks and with respect to the central BH attractor; therefore, in Section 3.1, the ℓcorotating pair of tori is addressed, while in Section 3.2 we focus our attention on the ℓcounterrotating case. We shall see that the results of Section 3.1 also apply to the description of ${\ell }$counterrotating tori in a Schwarzschild (static) spacetime. The double-torus disk system is characterized by the existence and stability conditions. We consider first all possible states for the pair of accretion disks with fixed topology, and then we concentrate on their evolutions. We will prove that some configurations are prohibited. Then we narrow the space of the system parameters to specific regions according to the dimensionless BH spin. The case of ℓcounterrotating couples around a rotating attractor is in fact much more articulated in comparison to the ℓcorotating (or ℓcounterrotating tori orbiting a Schwarzschild black hole) case. This case is hugely diverse for the classes of attractors and for the disk spin orientation with respect to the central attractor. Therefore, a different approach adapted for the diversity of cases is required. In order to better analyze the situation, we have split the analysis into two sections, Sections 3.2.1 and 3.2.2; in the first we consider the case in which the inner torus of the couple is counterrotating with respect to the attractor, and then we address the inner corotating torus. Accurate analysis in the parameter space also allows us to discuss the possible and forbidden lines of evolution for a fixed couple. We close Section 3 with Section 3.3, where the possibilities of collision between tori and of tori merging are considered. We investigate the conditions for collision occurrence, drawing a description of the associated unstable macro-configurations. Both ℓcorotaing and ℓcounterrotating cases are addressed. We discuss mechanisms that may lead to tori colliding according to our model prescription. This section also refers the reader to Appendix A.1, where further details are provided. Indications on possible observational evidence of double-torus disks and their evolution are provided in brief, in Section 4. We close this article in Section 5 with a summary and brief discussion of future prospecs.  Appendices A and B follow.

The Kerr metric line element in the Boyer–Lindquist (BL) coordinates $\{t,r,\theta ,\phi \}$ reads

$$\begin{eqnarray}&&{{ds}}^{2}=-{{dt}}^{2}+\displaystyle \frac{{\rho }^{2}}{{\rm{\Delta }}}{{dr}}^{2}+{\rho }^{2}d{\theta }^{2}+({r}^{2}+{a}^{2}){\sin }^{2}\theta d{\phi }^{2}\\ &&\,+\displaystyle \frac{2M}{{\rho }^{2}}r{({dt}-a{\sin }^{2}\theta d\phi )}^{2},\\ &&\quad {\rm{where}}\quad {\rho }^{2}\equiv {r}^{2}+{a}^{2}\cos {\theta }^{2}\quad {\rm{and}}\,{\rm{}}\\ &&\quad {\rm{\Delta }}\equiv {r}^{2}-2{Mr}+{a}^{2},\end{eqnarray} \tag{ 1 }$$

where $a=J/M\in [0,M]$ is the specific angular momentum, J is the total angular momentum of the gravitational source, and M is the gravitational mass parameter. The horizons ${r}_{-}\lt {r}_{+}$ and the outer static limit ${r}_{\epsilon }^{+}$ are respectively given by¹

$$\begin{eqnarray}&&{r}_{\pm }\equiv M\pm \sqrt{{M}^{2}-{a}^{2}};\quad {r}_{\epsilon }^{+}\equiv M+\sqrt{{M}^{2}-{a}^{2}\cos {\theta }^{2}};\end{eqnarray} \tag{ 2 }$$

where ${r}_{+}\lt {r}_{\epsilon }^{+}$ on $\theta \ne 0$ and ${r}_{\epsilon }^{+}=2M$ in the equatorial plane $\theta =\pi /2$. The non-rotating limiting case a = 0 is the Schwarzschild metric while the extreme Kerr BH has dimensionless spin $a/M=1$. In the Kerr geometry, the quantities

$$\begin{eqnarray}&&E\equiv -{g}_{\alpha \beta }{\xi }_{t}^{\alpha }{p}^{\beta }=-{p}_{t},\quad L\equiv {g}_{\alpha \beta }{\xi }_{\phi }^{\alpha }{p}^{\beta }={p}_{\phi },\end{eqnarray} \tag{ 3 }$$

are constants of motion, where ${\xi }_{\phi }={\partial }_{\phi }$ is the rotational Killing field, ${\xi }_{t}={\partial }_{t}$ is the Killing field representing the stationarity of the spacetime, and ${p}^{\alpha }$ is the particle four-momentum. The constant L in Equation (3) may be interpreted as the axial component of the angular momentum of a test particle following time-like geodesics and E is representing the total energy of the test particle coming from radial infinity, as measured by a static observer at infinity. Due to the symmetries of the metric tensor (1), the test particle dynamics is invariant under the mutual transformation of the parameters $(a,L)\to (-a,-L)$, and we could restrict the analysis of the test particle's circular motion to the case of positive values of a for corotating $(L\gt 0)$ and counterrotating $(L\lt 0)$ orbits.

In this work, we particularize our analysis to toroidal configurations of a perfect fluid orbiting a Kerr BH attractor. The energy–momentum tensor for a one-species particle perfect fluid system is described by

$$\begin{eqnarray}&&{T}_{\alpha \beta }=(\varrho +p){u}_{\alpha }{u}_{\beta }+\ {{pg}}_{\alpha \beta },\end{eqnarray} \tag{ 4 }$$

where ${u}^{\alpha }$ is a time-like flow vector field and ϱ and p are the total energy density and pressure, respectively, as measured by an observer comoving with the fluid with velocity ${u}^{\alpha }$. For the symmetries of the problem, we assume ${\partial }_{t}{\boldsymbol{Q}}=0$ and ${\partial }_{\varphi }{\boldsymbol{Q}}=0$, with ${\boldsymbol{Q}}$ being a generic spacetime tensor. According to these assumptions, the continuity equation is identically satisfied and the fluid dynamics is governed by the Euler equation,

$$\begin{eqnarray}&&(p+\varrho ){u}^{\alpha }{{\rm{\nabla }}}_{\alpha }{u}^{\gamma }+\ {h}^{\beta \gamma }{{\rm{\nabla }}}_{\beta }p=0,\end{eqnarray} \tag{ 5 }$$

where ${{\rm{\nabla }}}_{\alpha }{g}_{\beta \gamma }=0$ and ${h}_{\alpha \beta }={g}_{\alpha \beta }+{u}_{\alpha }{u}_{\beta }$ is the projection tensor (Pugliese & Kroon 2012; Pugliese & Montani 2015). Assuming a barotropic equation of state $p=p(\varrho )$, and orbital motion with ${u}^{\theta }=0$ and u^r = 0, Equation (5) implies

$$\begin{eqnarray}\displaystyle \,\frac{{\partial }_{\mu }p}{\varrho +p} & = & -{\partial }_{\mu }W+\displaystyle \frac{{\rm{\Omega }}{\partial }_{\mu }{\ell }}{1-{\rm{\Omega }}{\ell }},\quad {\ell }\equiv \displaystyle \frac{L}{E},\\ W & \equiv & \mathrm{ln}{V}_{\mathrm{eff}}({\ell }),\ {V}_{\mathrm{eff}}({\ell })={u}_{t}=\pm \sqrt{\displaystyle \frac{{g}_{\phi t}^{2}-{g}_{{tt}}{g}_{\phi \phi }}{{g}_{\phi \phi }+2{\ell }{g}_{\phi t}+{{\ell }}^{2}{g}_{{tt}}}},\end{eqnarray} \tag{ 6 }$$

where ${\rm{\Omega }}={u}^{\phi }/{u}^{t}$ is the relativistic angular frequency of the fluid relative to the distant observer, and the Paczyński-Wiita (P-W) potential $W(r;{\ell },a)$ and the effective potential for the fluid ${V}_{\mathrm{eff}}(r;{\ell },a)$ are introduced. These functions of position reflect the background Kerr geometry through the parameter a, and the centrifugal effects through the fluid-specific angular momenta ℓ, here assumed constant and conserved (see also Lei et al. 2008; Abramowicz 2008). A natural extremal limit on the extension of both corotating and counterrotating tori occurs due to the cosmic repulsion at the so-called static radius that is independent of the BH spin (Stuchlik 1983, 2005; Stuchlik & Hledik 1999; Stuchlík et al. 2000; Slaný & Stuchlík 2005; Stuchlik et al. 2004; Stuchlik et al. 2009).

The effective potential in Equation (6) is invariant under the mutual transformation of the parameters $(a,{\ell })\to (-a,-{\ell })$. Therefore, analogously to the analysis of test particle dynamics, we can assume $a\gt 0$ and consider ${\ell }\gt 0$ for corotating and ${\ell }\lt 0$ for counterrotating fluids within the notation $(\mp )$.

The ringed accretion disks, introduced in Pugliese & Montani (2015) and Pugliese & Stuchlík (2015, 2016a), represent a fully general relativistic model of toroidal disk configurations ${{\boldsymbol{C}}}^{n}={\bigcup }^{n}{C}_{i}$, consisting of a collection of n subconfigurations (configuration order n) of corotating and counterrotating toroidal rings orbiting a supermassive Kerr attractor—see Figure 3. Since tori can be corotating or counterrotating with respect to the BH, assuming first a couple $({C}_{a},{C}_{b})$, orbiting in the equatorial plane of a given Kerr BH with specific angular momentum $({{\ell }}_{a},{{\ell }}_{b})$. We need to introduce the concept of ℓcorotating disks, defined by the condition ${{\ell }}_{a}{{\ell }}_{b}\gt 0$, and ℓcounterrotating disks, defined by the relations ${{\ell }}_{a}{{\ell }}_{b}\lt 0$. The two ℓcorotating tori can be both corotating, ${\ell }a\gt 0$, or counterrotating, ${\ell }a\lt 0$, with respect to the central attractor—see Figure 4.

The construction of the ringed configurations is actually independent of the adopted model for the single accretion disk (subconfiguration or ring). However, to simplify the discussion, we consider here each toroid of the ringed disk governed by the general relativistic hydrodynamic Boyer condition of equilibrium configurations of rotating perfect fluids. We will see that in situations where the curvature effects of the Kerr geometry are significant, the results are largely independent of the specific characteristics of the model for the single disk configuration, which are primarily based on the characteristics of the geodesic structure of the Kerr spacetime related to the matter distribution. This is a geometric property consisting of the union of the orbital regions with boundaries at the notable radii ${{\boldsymbol{R}}}_{{\rm{N}}}^{\pm }\equiv \{{r}_{\gamma }^{\pm },{r}_{\mathrm{mbo}}^{\pm },{r}_{\mathrm{mso}}^{\pm }\}$. It can be decomposed, for $a\ne 0$, into ${{\boldsymbol{R}}}_{{\rm{N}}}^{-}$ for the corotating and ${{\boldsymbol{R}}}_{{\rm{N}}}^{+}$ for counterrotating matter. Specifically, for time-like circular geodetical particle orbits, ${r}_{\gamma }^{\pm }$ is the marginally circular orbit or the photon circular orbit; time-like circular orbits can fill the spacetime region $r\gt {r}_{\gamma }^{\pm }$. For the marginally stable circular orbit ${r}_{\mathrm{mso}}^{\pm }$: stable orbits are in $r\gt {r}_{\mathrm{mso}}^{\pm }$ for counterrotating and corotating particles, respectively. The marginally bounded circular orbit is ${r}_{\mathrm{mbo}}^{\pm }$, where ${E}_{\pm }({r}_{\mathrm{mbo}}^{\pm })=1$ (Stuchlik 1981a, 1981b; Stuchlik & Slany 2004; Stuchlik & Kotrlova 2009; Pugliese et al. 2011a, 2011b, 2013; Pugliese & Quevedo 2015)—see Figures 1 and 2. Given ${r}_{i}\in { \mathcal R }$, we adopt the following notation for any function ${\boldsymbol{Q}}(r):\,{{\boldsymbol{Q}}}_{i}\equiv {\boldsymbol{Q}}({r}_{i})$, for example, ${{\ell }}_{\mathrm{mso}}^{+}\equiv {{\ell }}_{+}({r}_{\mathrm{mso}}^{+})$ and more generally, given the radius r_• and the function ${\boldsymbol{Q}}(r)$, there is ${{\boldsymbol{Q}}}_{\bullet }\equiv {\boldsymbol{Q}}({r}_{\bullet })$. Since the intersection set of ${r}_{{\rm{N}}}^{\pm }$ is not empty, the character of the geodesic structure will be particularly relevant in the characterization of the ℓcounterrotating sequences (Pugliese & Stuchlík 2015).

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According to the Boyer theory on equipressure surfaces applied to a P-D torus, the toroidal surfaces are the equipotential surfaces of the effective potential ${V}_{\mathrm{eff}}({\ell },r)$, which are solutions of ${V}_{\mathrm{eff}}=K\,=$ constant or $\mathrm{ln}({V}_{\mathrm{eff}})={\rm{c}}=\mathrm{constant}$ (Boyer 1965; Kozlowski et al. 1978). These also correspond to surfaces of constant density, specific angular momentum ℓ, and constant relativistic angular frequency Ω, where ${\rm{\Omega }}={\rm{\Omega }}({\ell })$, as a consequence of the von Zeipel theorem (Abramowicz 1971; Kozlowski et al. 1978; Zanotti & Pugliese 2015). Then, each Boyer surface is uniquely identified by the couple of parameters ${\boldsymbol{p}}\equiv ({\ell },K)$. We focus on the solution of Equation (6), W = constant, associated with the critical points of the effective potential, assuming constant specific angular momentum and parameter K. Considering ${{\rm{\Delta }}}_{\mathrm{crit}}\equiv [{r}_{\max },{r}_{\min }]$, whose boundaries correspond to the maximum and minimum points of the effective potential, respectively, we have that the centers r_cent of the closed configurations ${C}_{\pm }$ are located at the minimum points ${r}_{\min }\gt {r}_{\mathrm{mso}}^{\pm }$ of the effective potential, where the hydrostatic pressure reaches a maximum. The toroidal surfaces are characterized by ${K}_{\pm }\,\in [{K}_{\min }^{\pm },{K}_{\max }^{\pm }[\subset ]{K}_{\mathrm{mso}}^{\pm },1[\,\equiv \,{\boldsymbol{K}}0$ and momentum ${{\ell }}_{\pm }\lessgtr {{\ell }}_{\mathrm{mso}}^{\pm }\lessgtr 0$ respectively. The inner edge of the Boyer surface is at ${r}_{\mathrm{in}}\in {{\rm{\Delta }}}_{\mathrm{crit}}$, or ${r}_{\mathrm{in}}\equiv {y}_{3}$ on the equatorial plane; the outer edge is at ${r}_{\mathrm{out}}\gt {r}_{\min }$, or ${r}_{\mathrm{out}}\equiv {y}_{1}$ on the equatorial plane as in Figure 3. A further matter configuration closest to the BH is at ${r}_{\mathrm{in}}\lt {r}_{\max }$. The limiting case of ${K}_{\pm }={K}_{\min }^{\pm }$ corresponds to a one-dimensional ring of matter located in ${r}_{\min }^{\pm }$. Equilibrium configurations, with topology $C$, exist for $\pm {{\ell }}_{\mp }\gt \pm {{\ell }}_{\mathrm{mso}}^{\mp }$ centered on $r\gt {r}_{\mathrm{mso}}^{\mp }$. In general, we denote by the label $(i)$, with $i\in \{1,2,3\}$, any quantity ${\boldsymbol{Q}}$ related to the range of specific angular momentum ${\boldsymbol{Li}}$, for example, ${C}_{1}^{+}$ indicates a closed regular counterrotating configuration with specific angular momentum ${{\ell }}_{1}^{+}\in {\boldsymbol{L}}{1}^{+}$.

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The local maxima of the effective potential r_max correspond to points of minimum hydrostatic pressure and the P-W points of gravitational and hydrostatic instability. No maxima of the effective potential exist for $\pm {{\ell }}_{\mp }\gt {{\ell }}_{\gamma }^{\pm }$ (${\boldsymbol{L}}{3}^{\mp }$); therefore, only equilibrium configurations ${C}_{3}$ are possible. An accretion overflow of matter from the closed, cusped configurations in ${C}_{\times }^{\pm }$ (see Figure 3) toward the attractor can occur from the instability point ${r}_{\times }^{\pm }\equiv {r}_{\max }\in ]{r}_{\mathrm{mbo}}^{\pm },{r}_{\mathrm{mso}}^{\pm }[$, if ${K}_{\max }\in {\boldsymbol{K}}{0}^{\pm }$ with specific angular momentum ${\ell }\in \,]{{\ell }}_{\mathrm{mbo}}^{+},{{\ell }}_{\mathrm{mso}}^{+}[\,\equiv \,{\boldsymbol{L}}{1}^{+}$ or ${\ell }\in ]{{\ell }}_{\mathrm{mso}}^{-},{{\ell }}_{\mathrm{mbo}}^{-}[\,\equiv \,{\boldsymbol{L}}{1}^{-}$. Otherwise, there can be funnels of material along an open configuration ${{\rm{O}}}_{\times }^{\pm }$, proto-jets, or for brevity, jets, which represent limiting topologies for the closed surfaces (Kozlowski et al. 1978; Sikora 1981; Madau 1988; Lyutikov 2009; Lasota et al. 2016; Sadowski et al. 2016) with ${K}_{\max }^{\pm }\geqslant 1$ (${\boldsymbol{K}}{1}^{\pm }$), "launched" from the point ${r}_{{\rm{J}}}^{\pm }\equiv {r}_{\max }\in ]{r}_{\gamma }^{\pm },{r}_{\mathrm{mbo}}^{\pm }]$ with specific angular momentum ${\ell }\in ]{{\ell }}_{\gamma }^{+},{{\ell }}_{\mathrm{mbo}}^{+}[\,\equiv \,{\boldsymbol{L}}{2}^{+}$ or $]{{\ell }}_{\mathrm{mbo}}^{-},{{\ell }}_{\gamma }^{-}[\,\equiv \,{\boldsymbol{L}}{2}^{-}$. However, we can more precisely locate the points of maximum pressure, which correspond to the center of each torus at ${r}_{\min }^{\pm }\gt {r}_{\mathrm{mso}}^{\pm },$  by introducing the "complementary" geodesic structure associated with the geodesic structure constituted by the notable radii ${{\boldsymbol{R}}}_{{\rm{N}}}$ and by defining the radii ${\bar{{\mathfrak{r}}}}_{{\rm{N}}}\in {\bar{{\boldsymbol{R}}}}_{{\rm{N}}}:$ ${\bar{{\mathfrak{r}}}}_{{\rm{N}}}\gt {r}_{{\rm{N}}}$ solutions of ${\bar{{\ell }}}_{{\rm{N}}}\equiv {\ell }({\bar{{\mathfrak{r}}}}_{{\rm{N}}})={\ell }({r}_{{\rm{N}}})\equiv {{\ell }}_{{\rm{N}}}$—see Figures 1 and 2. These radii satisfy the same equations as the notable radii ${r}_{{\rm{N}}}\in {{\boldsymbol{R}}}_{{\rm{N}}}$ for corotating and counterrotating configurations, analogously to the couples ${r}_{{ \mathcal M }}^{\pm }$ and ${\bar{{\mathfrak{r}}}}_{{ \mathcal M }}^{\pm }$, where ${r}_{{ \mathcal M }}^{\pm }\gt {r}_{\mathrm{mso}}^{\pm }$ and the associated ${{\ell }}_{{ \mathcal M }}^{\pm }$ is a maximum of ${\partial }_{r}| {\ell }(r)|$ (Pugliese & Stuchlík 2016a). The geodesic structure of spacetime and the complementary geodesic structure are both significant in the analysis, especially in the case of ℓcounterrotating couples. Since ${r}_{\gamma }^{\pm }\lt {r}_{\mathrm{mbo}}^{\pm }\lt {r}_{\mathrm{mso}}^{\pm }\lt {\bar{{\mathfrak{r}}}}_{\mathrm{mbo}}^{\pm }\lt {\bar{{\mathfrak{r}}}}_{\gamma }^{\pm }$, the location of the radii ${r}_{{ \mathcal M }}$ and ${\bar{{\mathfrak{r}}}}_{{ \mathcal M }}$ depends on the rotation with respect to the Kerr attractor. Clearly, the marginally stable orbit ${r}_{\mathrm{mso}}$ is the only solution of ${r}_{{\rm{N}}}={\bar{{\mathfrak{r}}}}_{{\rm{N}}}$. Thus, the configurations ${()}_{1}$ are centered on $]{r}_{\mathrm{mso}},{\bar{{\mathfrak{r}}}}_{\mathrm{mbo}}[$(with the accretion point in ${r}_{\times }\in ]{r}_{\mathrm{mbo}},{r}_{\mathrm{mso}}[$), the ${()}_{2}$ rings have centers in the range $[{\bar{{\mathfrak{r}}}}_{\mathrm{mbo}},{\bar{{\mathfrak{r}}}}_{\gamma }[$ (with ${r}_{{\rm{J}}}\in \,]{r}_{\gamma },{r}_{\mathrm{mbo}}[$), and finally the ${C}_{3}$ disks are centered at $r\geqslant {\bar{{\mathfrak{r}}}}_{\gamma }$.

However, global instability of the entire macro-configuration may be associated with two distinct models of an unstable ringed torus with a degenerate topology. Related to these, there are two types of instabilities emerging in an orbiting macro-structure. First, there is the emergence of a P-W instability in one of its ring and collision among the subconfigurations. The P-W local instability affects one or more rings of the ringed disk decomposition, and it can then destabilize the macro-configuration when the rings are no longer separated and feeding (overlapping) of material occurs. Second, contact (or geometrical correlation) in this model causes the collision and penetration of matter, eventually leading to the feeding of one subconfiguration with material and specific angular momentum supplied to another consecutive ring of the decomposition. This mechanism could possibly end in the change of the ringed disk morphology and topology. Accordingly, there is a macro-structure ${{\boldsymbol{C}}}_{\odot }^{{\boldsymbol{n}}}$, with a number ${\mathfrak{r}}\in [0,n-1]$ of contact points between the boundaries of two consecutive rings (rank of ${{\boldsymbol{C}}}_{\odot }^{{\boldsymbol{n}}}$), and a macro-structure ${C}_{\times }^{{\boldsymbol{n}}}$, with ${{\mathfrak{r}}}_{\times }\in [0,n]$ instability P-W points. The number ${{\mathfrak{r}}}_{\times }$ is called the rank of the ringed disk ${C}_{\times }^{{\boldsymbol{n}}}$. Finally, we have the macro-structure ${C}_{\odot }^{{\boldsymbol{x}}\,n}$, characterized by at least one contact point that is also an instability point.

If ${{\mathfrak{r}}}_{\times }=1$ and the inner ring ${{\boldsymbol{C}}}_{\times }^{1}$ of its decomposition is in accretion, then the entire ringed disk could be globally stable (Pugliese & Stuchlík 2015).

We shall describe the system made up of two tori in a Kerr geometry as ringed accretion disks ${{\boldsymbol{C}}}^{2}$ of the order n = 2 (state)—Figure (4). We can introduce the elongation ${{\rm{\Lambda }}}_{{{\boldsymbol{C}}}^{2}}$ of ${{\boldsymbol{C}}}^{2}:\,{C}_{a}\lt {C}_{b}$ and the spacing ${\bar{{\rm{\Lambda }}}}_{\mathrm{2,1}}\equiv [{y}_{1}^{a},{y}_{3}^{b}]$  by the relations

$$\begin{eqnarray}{{\rm{\Lambda }}}_{{{\boldsymbol{C}}}^{2}} & \equiv & [{y}_{3}^{a},{y}_{1}^{b}]=\left(\bigcup _{i=1}^{2}{{\rm{\Lambda }}}_{i}\right)+{\bar{{\rm{\Lambda }}}}_{\mathrm{2,1}},\\ {\lambda }_{{{\boldsymbol{C}}}^{2}} & \equiv & {y}_{1}^{b}-{y}_{3}^{a}=\displaystyle \sum _{i}^{2}{\lambda }_{i}+{\bar{\lambda }}_{\mathrm{2,1}}\geqslant {\lambda }_{{{\boldsymbol{C}}}^{2}}^{\inf }\equiv {\displaystyle \sum _{i}^{2}{\lambda }_{i}|}_{{\displaystyle \sum }_{i}{\lambda }_{i}},\end{eqnarray} \tag{ 7 }$$

where ${\bar{{\rm{\Lambda }}}}_{i}$ and ${{\rm{\Lambda }}}_{i}$ are the spacing and elongation of each ring and ${\lambda }_{{{\boldsymbol{C}}}^{2}}$ is the measure of the elongation of the (separated) configuration ${{\boldsymbol{C}}}^{2}$—see Figure 3. Equation (7) shows that the minimum value ${\lambda }_{{{\boldsymbol{C}}}^{2}}^{\inf }$ of the elongation ${\lambda }_{{{\boldsymbol{C}}}^{2}}$ is achieved, at fixed ${\sum }_{i}^{n}{\lambda }_{i}$, when ${\bar{\lambda }}_{\mathrm{2,1}}=0$, that is, for a ${{\boldsymbol{C}}}_{\odot }^{2}$ configuration of rank ${\mathfrak{r}}={{\mathfrak{r}}}_{\max }$. As demostrated in Pugliese & Stuchlík (2015), we can introduce the effective potential ${{V}_{\mathrm{eff}}^{{{\boldsymbol{C}}}^{2}}| }_{{K}_{i}}$ of the decomposed ${{\boldsymbol{C}}}^{n}$ macro-structure and the effective potential ${V}_{\mathrm{eff}}^{{{\boldsymbol{C}}}^{2}}$ of the configuration:

$$\begin{eqnarray}{{V}_{\mathrm{eff}}^{{{\boldsymbol{C}}}^{2}}| }_{{K}_{i}} & \equiv & {V}_{\mathrm{eff}}^{1}{\rm{\Theta }}(-{K}_{1})\,\bigcup \,{V}_{\mathrm{eff}}^{2}\,{\rm{\Theta }}\,(-{K}_{2})\quad {\rm{and}}\\ {V}_{\mathrm{eff}}^{{{\boldsymbol{C}}}^{2}} & \equiv & {V}_{\mathrm{eff}}^{i}({{\ell }}_{i})\,{\rm{\Theta }}\,({r}_{\min }^{o}-r)\,{\rm{\Theta }}\,(r-{r}_{+})\,\bigcup \,{V}_{\mathrm{eff}}^{o}({{\ell }}_{o})\,{\rm{\Theta }}\,(r-{r}_{\min }^{i}),\end{eqnarray} \tag{ 8 }$$

where ${\rm{\Theta }}(-{K}_{i})$ is the Heaviside (step) function such that ${\rm{\Theta }}(-{K}_{i})=1$ for ${V}_{\mathrm{eff}}^{i}\lt {K}_{i}$ and ${\rm{\Theta }}(-{K}_{i})=0$ for ${V}_{\mathrm{eff}}^{i}\gt {K}_{i}$, so that the curve ${V}_{\mathrm{eff}}^{{\boldsymbol{C}}}(r)$ is the union of all curves ${V}_{\mathrm{eff}}^{i}(r)\lt {K}_{i}$ of its decomposition. The potential ${{V}_{\mathrm{eff}}^{{{\boldsymbol{C}}}^{2}}| }_{{K}_{i}}$ regulates the behavior of each ring, taking into account the gravitational effects induced by the background and the centrifugal effect induced by the motion of the fluid, while the potential ${V}_{\mathrm{eff}}^{{{\boldsymbol{C}}}^{2}}$ governs the individual configurations considered as part of the macro-configuration—see Figure 3. Details on the effective potential, definition of the differential rotation of the decomposition, specific angular momentum of the ringed disk, and also thickness of the ringed disk can be found in Pugliese & Stuchlík (2015), where these configurations were first introduced, and then detailed in Pugliese & Stuchlík (2016a) for configuration order $n\geqslant 2$. Here, we particularize the introduced concepts to the case of only two rings. In Section 3 we characterize the double accretion disk system, focusing in Section 3.1 on the ℓcorotating couples, while in Section 3.2 we discuss the case of ℓcounterrotating couples.

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To simplify and illustrate the discussion, we use special graphs representing a couple of accretion disks and their evolution within the constraints to which they are subjected. The case of a couple of tori orbiting around a single central Kerr BH involves in general a remarkably large number of possible configurations: for a couple with a fixed and equal critical topology, there could be n = 8 different states according to their rotation and relative position of the centers. The couple $({C}_{\times },{{\rm{O}}}_{\times })$, with a different but fixed topology, could be in n = 16 different states, while for the state ${C}_{i}-{()}_{\times }$, with one equilibrium topology, we need to address n = 48 different cases—see Figure 5 for a sample of cases.

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The use of graphic schemes is crucial for the representation of these cases to quickly collect the different constraints on the existence and evolution of the states and for reference in our discussion. Therefore, although the following analysis is quite independent from the graph formalism, for easy reference we include here a brief description of this formalism and a discussion of the graph construction, introducing the essential blocks composing the graphs used in this work, and the list of notations and basic concepts related to these structures. We refer to Appendix A for details on the construction and interpretation of graphs associated with these systems, while in Figure 6 we present the main blocks the graphs are made of, with a brief description that also provides a list of the main notation and definitions used throughout this work.

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List of principal notations in the graph construction with reference to Figure 6. A graph vertex represents one configuration of the torus couple as defined by the ringed disk topology and fluid rotation with respect to the central Kerr BH attractor; a vertex then stands for one configuration of the set ${()}^{\pm }=\{{C}^{\pm },{C}_{\times }^{\pm },{{\rm{O}}}_{\times }^{\pm }\}$. The state lines connect two vertexes of the graph and represent a fixed couple of accretion torus disks. A monochromatic graph has one monochromatic state, i.e., a state line connecting two ℓcorotating configurations. A bichromatic graph has one bichromatic states, i.e., a state line connecting two ℓcounterrotating configurations. For configuration sequentiality, which is signed on a state line and associated with the notation $\lt$ or $\gt$, we intend this to be the ordered sequence of the maximum points of the pressure, or r_min, the minimum of the effective potential which corresponds to the configuration centers. Therefore, in relation to a couple of rings, the terms "internal" (inner-i) or "external" (outer-o) will always refer, unless otherwise specified, to the sequence ordered according to the center location. For critical sequentiality, attached to a state line and associated with the symbols $\succ$ and $\prec$, we refer to the sequentiality according to the location of the minimum points of the pressure, or r_max, the maximum point of the effective potential (in ${\boldsymbol{L}}1$ or ${\boldsymbol{L}}2$). A state line is completely oriented if both the configuration and critical sequentiality are specified, where the last one may be defined. Two configurations are correlated if they can be in contact, which implies collision in accordance with the constraints. In some cases, there are particularly restrictive conditions to be satisfied for a correlation to occur (constrained non-correlation). The addition of specific information on the lines and vertices of the graph, for example, the color of the correlation and sequentiality, is called graph decoration. An evolutive line connects two vertexes of two different state lines of the graph, and it represents the evolution of one configuration from one (starting) topology (vertex) to another topology (a vertex of a different state), for example from a $C$ configuration to a ${C}_{\times }$ in accretion. Evolutive lines may be composed to be closed on an initial vertex of initial state lines, creating a loop—see Appendix A.1. The central state of the graph is the couple, with the graph configurations describing the evolution toward different states (every evolutive line starts, ends, or passes through the central state). In this work, the central state is the initial state line according to the evolution signaled by the evolutive lines. Further details can be found in Appendix A. State lines for ℓcorotating couples are given in Figure 13 and state lines for ℓcounterrotating couples are in Figure 14. Figure 8 describes monochromatic graphs while Figure 9 shows bichromatic graphs.

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In this section, we discuss the existence and stability of the ringed disk ${{\boldsymbol{C}}}^{2}$ of order n = 2, made up of two toroidal configurations orbiting a spinning BH attractor.

We first consider all possible states for the couple of accretion disks with fixed topology. In the graph formalism, the analysis represents the research on all of the possible state and evolutive lines and their decorations (see Figure 6 and the end of Section 2) according to the separation constraint.² We refer to Appendix A for details on the construction and interpretation of the graphs associated with these systems.

We shall prove that some states, or some decorations for a state, are prohibited by several conditions, determined mainly by the dimensionless spin of the attractor and by the separation constraint. Specifically, we discuss the evolution of the configurations toward the phase of accretion onto the attractor, which could lead to the violation of the separation condition. We study the collisions between the rings of the couple, which gives rise to the emergence of the ${{\boldsymbol{C}}}_{\odot }^{2}$ (critical) macro-configuration, eventually causing the rings to merge.

The states could be further constrained by the maximum possible extension of the closed configurations for fixed angular momentum, defined by the supremum of the parameter K, $\sup K$. It is clear that for the ${C}_{1}$ configurations, we should consider the maximum of the elongation at the accretion ${\lambda }_{\times }$ and for the ${C}_{2}$ disks, the superior for ${K}_{\max }=1$. On the other hand, there is no similar constraint for the ${C}_{3}$ configurations since there are no minimum points for the hydrostatic pressure. However, we can infer the presence of the constraints in terms of the location of the inner and outer edges of the torus with respect to the notable radii by considering the results of Pugliese & Stuchlík (2016a).

In Section 3.1, we will show how a monochromatic graph, generally describing the situation for an ℓcorotating couple in any Kerr spacetime with $a\in [0,M]$, also describes the states and evolution of an ℓcounterrotating couple orbiting a Schwarzschild attractor (a = 0), due to the particular geodesic structure of this static spacetime. Figure 13 shows the possible state lines for the ℓcorotating couples, while the possible state lines for the ℓcounterrotating couples in a Kerr spacetime are listed in Figure 14. Table 2 also provides guidance on the sequentiality of the ℓcounterrotating couples according to criticality and configuration order. The decorations of the state lines generally show the emergence of possible collisions in accordance with the criteria used in the construction of the table, the location of the tori, and the possible relation between the critical points.

Restricting our study to ${{\boldsymbol{C}}}^{2}$ configurations, we concentrate our attention on the classification of the configurations with specific angular momenta ${\ell }\in {\boldsymbol{Li}}$ with $i\in \{1,2\}$ (Pugliese & Stuchlík 2015). Some of these ringed disks are constrained to a configuration order ${n}_{\max }=2$.

• 1.  
  The configuration

  $$\begin{eqnarray}&&{\bar{{\mathfrak{C}}}}_{0}\,:\,{{\rm{\Delta }}}_{\mathrm{cri}}^{i}\cap {{\rm{\Delta }}}_{\mathrm{cri}}^{o}=\varnothing ,\quad {\rm{there}}\,{\rm{is}}\\ &&\quad a\ne 0,\,{{\ell }}_{i}{{\ell }}_{o}\lt 0,\,({)}_{i}^{-}\lt ({)}_{o}^{+}\quad {\rm{with}}\\ &&\quad ({)}_{i}^{-}\,\prec \,({)}_{o}^{+},\quad {n}_{\max }({\bar{{\mathfrak{C}}}}_{0})=2,\end{eqnarray} \tag{ 9 }$$

  and we have

  $$\begin{eqnarray}&&-{{\ell }}_{+}^{o}\in ]-{{\ell }}_{\mathrm{mso}}^{+},-{{\ell }}_{+}({r}_{\mathrm{mso}}^{-})[\quad {\rm{and}}\\ &&{{\ell }}_{-}^{i}\in ]{{\ell }}_{\mathrm{mso}}^{-},{{\ell }}_{-}({r}_{\mathrm{mso}}^{+})[.\end{eqnarray} \tag{ 10 }$$

  $$\begin{eqnarray}&&{\rm{For}}\,{\rm{}}a\in ]0,{a}_{{\aleph }_{1}}[\,:\,({)}_{o}^{+}\ne {{\rm{O}}}_{\times }^{+}\quad {\rm{for}}\quad {r}_{\mathrm{mbo}}^{+}\lt {r}_{\mathrm{mso}}^{-}\\ &&\quad {\rm{where}}\quad {a}_{{\aleph }_{1}}\,:\,{r}_{\mathrm{mbo}}^{+}={r}_{\mathrm{mso}}^{-}.\end{eqnarray} \tag{ 11 }$$

  Table 1 lists and summarizes the main features of the spin values singled out by analysis. Equation (10) is fulfilled for the following topologies: at $a\lt {\tilde{a}}_{\aleph }$ for ${()}^{-}=({)}_{1}^{-}$ and at $a\lt {a}_{{\iota }_{a}}\lt {\tilde{a}}_{\aleph }$, there could be only $({)}_{1}^{-}\lt ({)}_{1}^{+}$—Figure 9. Then, in $[{a}_{{\iota }_{a}},{\tilde{a}}_{\aleph }]$, there are $({)}_{1}^{-}\lt ({)}_{2}^{+}$ and $({)}_{1}^{-}\lt ({)}_{1}^{+}$, whereas at $a\in [{\tilde{a}}_{\aleph },{a}_{{\gamma }_{+}}^{-}]$, there is also $({)}_{2}^{-}\lt ({)}_{2}^{+}$, and in $[{a}_{{\gamma }_{+}}^{-},{\breve{a}}_{\aleph }]$ there is also $({)}_{2}^{-}\lt ({)}_{3}^{+}$. Finally, for $a\gt {\breve{a}}_{\aleph }$, the couple $({)}_{3}^{-}\lt ({)}_{3}^{+}$ is also possible. These constraints, however, are not sufficient to fully characterize the couples ${()}^{-}\lt {()}^{+}$ as discussed in Section 3.2.2; in fact not all of the couples ${()}^{-}\lt {()}^{+}$ belong to the ${\bar{{\mathfrak{C}}}}_{0}$ class.
• 2.  
  The configuration

  $$\begin{eqnarray}&&\,{\bar{{\mathfrak{C}}}}_{1a}\,:\,{r}_{\max }^{o}\in {{\rm{\Delta }}}_{\mathrm{crit}}^{i},\quad {\rm{thus}}\\ &&\quad {{\ell }}_{i}{{\ell }}_{o}\lt 0\quad {({)}_{i}^{-}\lessgtr ({)}_{o}^{+}()}_{o}=!{C}_{o}\quad {n}_{\max }({\bar{{\mathfrak{C}}}}_{1a})=2\end{eqnarray} \tag{ 12 }$$

  $$\begin{eqnarray}&&{\rm{if}}\quad ({)}_{o}^{-}\lt ({)}_{o}^{+}\ :\quad {\rm{then}}\\ &&\quad ({)}_{i}^{-}\prec ({)}_{o}^{+}\quad ({)}_{o}^{+}\,=\,!{C}^{+}\quad {n}_{\max }({\bar{{\mathfrak{C}}}}_{1a})=2\end{eqnarray} \tag{ 13 }$$

  $$\begin{eqnarray}&&{\rm{if}}\quad ({)}_{i}^{+}\lt ({)}_{o}^{-}\ :\quad {\rm{then}}\\ &&\quad ({)}_{o}^{-}\,=\,!{C}^{-}\quad ({)}_{i}^{+}\prec ({)}_{o}^{-}\,:\quad (a\gtrapprox 0).\end{eqnarray} \tag{ 14 }$$

  Here, for any relation $\bowtie$ among two quantities, in $\bowtie !$ the intensifier $(!)$, a reinforcement of the relation, indicates that this is a necessary relation which is always satisfied.
• 3.  
  The configuration

  $$\begin{eqnarray}&&{\bar{{\mathfrak{C}}}}_{1b}\,:\quad {{\rm{\Delta }}}_{\mathrm{crit}}^{i}\subset {{\rm{\Delta }}}_{\mathrm{crit}}^{o}\quad {\rm{then}}\,{\rm{there}}\,{\rm{is}}\,{\rm{}}\\ &&\quad {{\ell }}_{i}{{\ell }}_{o}\lessgtr 0\,{()}_{i}\lt {()}_{o}\\ &&\quad {()}_{i}\succ {()}_{o}\quad {()}_{o}\,=\,!{C}_{o}\quad {n}_{\max }({\bar{{\mathfrak{C}}}}_{1b})=\infty .\end{eqnarray} \tag{ 15 }$$

  A special case of this class of ringed disks are the couples ${{\ell }}_{(i+1)/i}=-1$, which can have ${{\ell }}_{-}/-{{\ell }}_{+}\gtrless 1$.

Table 1.  Classes of Attractors

Spins and Classes of Attractors
${a}_{{\aleph }_{2}}\equiv 0.172564M\,:-{{\ell }}_{\mathrm{mso}}^{+}={{\ell }}_{\mathrm{mbo}}^{-}$	${a}_{\iota }\equiv 0.3137M\,:{r}_{\mathrm{mbo}}^{-}={r}_{\gamma }^{+}$−$({{\boldsymbol{A}}}_{\iota }^{\lessgtr })$	${a}_{{\iota }_{a}}\equiv 0.372583M\,:{r}_{\mathrm{mso}}^{-}={r}_{\mathrm{mbo}}^{+}$ −$({{\boldsymbol{A}}}_{{\iota }_{a}}^{\lessgtr })$
${a}_{{\aleph }_{1}}=0.382542M\,:{{\ell }}_{\gamma }^{-}=-{{\ell }}_{+}({r}_{\mathrm{mso}}^{-})$	${a}_{{\aleph }_{0}}\equiv 0.390781M\,:{{\ell }}_{\gamma }^{-}=-{{\ell }}_{\mathrm{mbo}}^{+}$	${\tilde{a}}_{\aleph }\approx 0.461854M\,:{{\ell }}_{-}({r}_{\mathrm{mso}}^{+})={{\ell }}_{\mathrm{mbo}}^{-}$
${a}_{{}_{u}}=0.474033M\,:\,{\bar{{\mathfrak{r}}}}_{\mathrm{mbo}}^{+}={\bar{{\mathfrak{r}}}}_{\gamma }^{-}$	${a}_{\aleph }\approx 0.5089M\,:-{{\ell }}_{\mathrm{mso}}^{+}={{\ell }}_{\gamma }^{-}$	${a}_{{\gamma }_{+}}^{-}\equiv 0.638285M\,:{r}_{\gamma }^{+}={r}_{\mathrm{mso}}^{-}$
${a}_{1}\approx 0.707107M\,:{r}_{\gamma }^{-}={r}_{\epsilon }^{+}$	${\breve{a}}_{\aleph }=0.73688M\,:{{\ell }}_{-}({r}_{\mathrm{mso}}^{+})={{\ell }}_{\gamma }^{-}$	${a}_{b}\approx 0.828427M\,:{r}_{\mathrm{mbo}}^{-}={r}_{\epsilon }^{+}$
${a}_{{ \mathcal M }}^{-}\equiv 0.934313M\,:{{\ell }}_{\gamma }^{-}={{\ell }}_{{ \mathcal M }}^{-}$	${a}_{2}\approx 0.942809M\,:{r}_{\mathrm{mso}}^{-}={r}_{\epsilon }^{+}$	$\breve{a}\equiv 0.969174M\,:{\breve{{\ell }}}^{-}={r}_{\gamma }^{-}$ $({\breve{A}}_{\lessgtr })$

Note.  In general, given a spin value a_• the classes ${{\boldsymbol{A}}}_{\bullet }^{\lessgtr }$ stands for the ranges $a\in [0,{a}_{\bullet }[$ and $a\in ]{a}_{\bullet },M]$, respectively. Some of these classes are given alongside the spins.

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We finally note that, in a Kerr spacetime $(a\ne 0)$, the chromaticity of the graphs is determined by the relative rotation of the disks together with the rotation with respect to the attractor. However, the situation is different in the case of a static limit for the attractor with a = 0, where monochromatic graphs also describe ℓcounterrotating (i.e., ${{\ell }}_{i}{{\ell }}_{o}\lt 0$) and ℓcorotating (i.e., ${{\ell }}_{i}{{\ell }}_{o}\gt 0$) couples. In fact, in the case of a Schwarzschild attractor (static spacetime), it is still possible to consider a bichromatic graph with an arbitrary choice of tori relative rotation ${{\ell }}_{i}{{\ell }}_{j}\lt 0$, but the spacetime geodesic structure is singled out by the properties of the Schwarzschild geometry, independently of the sign of the fluid angular momentum. Therefore, at all effects, this bichromatic graph must undergo analysis on the monochromatic graph. However, a major difference between a bichromatic graph where a = 0 and a monochromatic one occurs in the static spacetime for the ℓcounterrotating case due to the possible evolutive loops of the bichromatic vertices, where collision between tori with ℓcounterrotating angular momenta may occur. In the following section, Section 3.1, we discuss the particular case of Schwarzschild geometry and ℓcorotating couples in Kerr spacetimes, while ℓcounterrotating couples orbiting a Kerr attractor are analyzed in Section 3.2.

3.1. The ${\ell }{corotating}$ Couples in Kerr Spacetime and the Case of Schwarzschild Geometry

3.2. The ${\ell }{counterrotating}$ Couples

We consider ℓcounterrotating couples orbiting a Kerr attractor with dimensionless spin $a/M\in [0,1]$ corresponding to bichromatic graphs—see schemes III and IV of Figure 4. In comparison to ℓcorotating couples (monochromatic graph), or ℓcounterrotating tori orbiting a Schwarzschild BH, this case is complex, owing to the diverse classes of attractors and to the disk spin orientation with respect to the central attractor (Pugliese & Stuchlík 2016a). It is therefore necessary to consider separately the case ${C}^{+}\lt {C}^{-}$ (inner counterrotating torus and outer corotating torus), which will be discussed in Section 3.2.1, from the ${C}^{-}\lt {C}^{+}$ case (inner corotating ring and outer counterrotating torus), which will be investigated in Section 3.2.2. The graph formalism can significantly simplify the analysis of the evolution of this particular double system. Using the results of Pugliese & Stuchlík (2016a), we build Figure 14 by collecting the main states for the graph from Figure 9. The most relevant effect distinguishing these pairs from the ℓcorotating couples is that the (counterrotating) outer torus of the couple in general may undergo a P-W instability phase, with the emergence of the instability point eventually giving rise to feeding of material to its companion inner torus.

The double sequentiality according to the configuration and criticality indices, respectively, in several lines in Table 14 and states in Figure 9 is not specified as depends on the vertex decoration in terms of the angular momentum. In fact, as demonstrated in Pugliese & Stuchlík (2015, 2016a), the sequentiality of the centers of ℓcounterrotating couples in equilibrium does not necessarily constrain the critical sequentiality (${\ell }\notin {\boldsymbol{L}}3$). There are special cases where, at fixed ${{\ell }}_{i}{{\ell }}_{o}\lt 0$, with ${()}_{i}\lt {()}_{o}$, there can be ${()}_{i}\,\prec \,{()}_{o}$, which corresponds to ${\bar{{\mathfrak{C}}}}_{0}$ in Equation (9) (if there is ${C}^{-}\lt {C}^{+}$). Otherwise, it corresponds to $\bar{{{\mathfrak{C}}}_{{1}_{a}}}$ in Equation (12) within the conditions Equations (13) or (14). Conversely, there can be ${()}_{i}\succ {()}_{o}$ i.e., a couple of the $\bar{{{\mathfrak{C}}}_{{1}_{b}}}$ class, in Equation (15), which also includes ℓcorotating couples.

Then, the outer vertex of the $\bar{{{\mathfrak{C}}}_{{1}_{{\rm{a}}}}}$ and $\bar{{{\mathfrak{C}}}_{{1}_{{\rm{b}}}}}$ couples must be in equilibrium or destroyed: this means that before the outer torus reaches its unstable phase, the ringed disk will be destroyed by collision, prohibiting any subsequent evolutive lines.

Such a situation may be prevented if a change in criticality order occurs, which means a transition from a $\bar{{{\mathfrak{C}}}_{{1}_{{\rm{a}}}}}$ or $\bar{{{\mathfrak{C}}}_{{1}_{{\rm{b}}}}}$ class to a $\bar{{{\mathfrak{C}}}_{0}}$ class. However, as the inversion of the configuration sequentiality is not permitted, such a transition could happen only for couples ${()}^{-}\lt {()}^{+}$, detailed in Section 3.2.2. In fact, in the isolated disks–attractor systems, the evolution of the couple is strongly determined by the decoration of the initial state. However, the conditions for the occurrence of this class transition are very complex, depending on the relation between characteristic values of the specific angular momentum $\{\mp {{\ell }}_{\mathrm{mso}}^{\pm },\mp {{\ell }}^{\pm }({r}_{\mathrm{mso}}^{\mp })\}$ which determine the boundaries of the ranges in Equation (10)—see also Figure 1.

More generally, for ${\ell }\notin {\boldsymbol{L}}3$, we can discuss the state sequentiality according to the arguments presented in Pugliese & Stuchlík (2015, 2016a). We distinguish two cases according to the magnitude of the specific angular momentum.

• 1.  
  $$\begin{eqnarray}&&\begin{array}{l}\text{For}\ | {{\ell }}_{-}/{{\ell }}_{+}| \gt 1\quad \mathrm{there}\ \mathrm{must}\ \mathrm{be}\quad {\bar{{\mathfrak{C}}}}_{1b}\\ \text{from Equation}\ {(15){\rm{or}}()}^{+}\lt {C}^{-}\quad {\rm{and}}\quad {()}^{+}\succ {C}^{-}.\end{array}\end{eqnarray} \tag{ 16 }$$

  This case, analyzed in Section 3.2.1 and represented by scheme III of Figure 4, is described by the graph in the left panel of Figure 9. As confirmed by this graph, only the inner counterrotating ring can accrete onto the source. On the other hand, the condition ${()}^{+}\lt {C}^{-}$ does not necessarily imply the angular momentum relation in Equation (16). Moreover, the condition ${{\ell }}_{-}\gt -{{\ell }}_{\mathrm{mso}}^{+}$ implies strong constraints on the initial state of the outer corotating torus of the couple. In fact, due to the constraint $| {{\ell }}_{-}/{{\ell }}_{+}| \gt 1$, if the attractor belongs to the class $a\lt {a}_{{\aleph }_{2}}$ (where $a={a}_{{\aleph }_{2}}\,:\,-{{\ell }}_{\mathrm{mso}}^{+}={{\ell }}_{\mathrm{mbo}}^{-}$), the outer corotating torus can belong to one of the ranges ${\boldsymbol{Li}}$. For $a\in ]{a}_{{\aleph }_{2}},{a}_{\aleph }[$, the specific angular momentum of the outer corotating torus has to be in ${\boldsymbol{L}}2$ or ${\boldsymbol{L}}3$, and for $a\gt {a}_{{ \mathcal M }}$, the torus is centered at $r\gt {r}_{{ \mathcal M }}^{-}$. For faster attractors with $a\gt {a}_{\aleph }$ (at $a={a}_{\aleph }\,:\,-{{\ell }}_{\mathrm{mso}}^{+}={{\ell }}_{\gamma }^{-}$), the corotating torus has to be located far from the attractor, as its specific angular momentum is in ${\boldsymbol{L}}3;$ the corresponding effective potential has thus no maximum points—Figure 1.
• 2.  
  $$\begin{eqnarray}&&\ \ \ \text{For}\ | {{\ell }}_{-}/{{\ell }}_{+}| \lt 1\,{\rm{there}}\,{\rm{is}}\,{\rm{}}\quad {()}^{-}\lessgtr {()}^{+},\,{\rm{and}}\quad {r}_{\min }^{-}\lt \bar{{\mathfrak{r}}}\\ &&\quad {\rm{where}}\,{\rm{}}\bar{{\mathfrak{r}}}\gt {r}_{\mathrm{mso}}^{-}\,:\,{{\ell }}_{-}=-{{\ell }}_{+}\gt {{\ell }}_{\mathrm{mso}}^{+}\end{eqnarray} \tag{ 17 }$$

  $$\begin{eqnarray}&&\,{\rm{if}}\quad {{\ell }}_{-}\in ]{{\ell }}_{-}{({r}_{\min }^{+}),{{\ell }}_{-}{(\bar{{\mathfrak{r}}})[{\rm{then}}{\rm{there}}{\rm{is}}{\rm{}}()}^{+}\lt ()}^{-},\end{eqnarray} \tag{ 18 }$$

  $$\begin{eqnarray}&&\,{\rm{if}}\quad {{\ell }}_{-}\in ]{{\ell }}_{\mathrm{mso}}^{-},{{\ell }}_{-}{({r}_{\min }^{+})[{\rm{then}}{\rm{there}}{\rm{is}}{\rm{}}()}^{-}\lt {()}^{+}.\end{eqnarray} \tag{ 19 }$$

  The sequentiality according to the criticality has been fixed in the first column of Table 2, which combines the additional restrictions provided by the angular momentum and the constraints from the complementary geodesic structure of spacetime ${\bar{{\boldsymbol{R}}}}_{{\rm{N}}}$, represented in Figure 1.The case ${()}^{+}\lt {()}^{-}$ in Equation (18) is illustrated in Figure 9, right panel, and represented in scheme IV of Figure 4; see also Figure 3. We note that in general, a small range of angular momentum in the case ${()}^{+}\lt {()}^{-}$ with $| {{\ell }}_{-}/{{\ell }}_{+}| \lt 1$ is associated with a limited orbital region which decreases as the torus distance from the attractor increases, or the attractor's dimensionless spin decreases, i.e., in the $R\equiv r/a\gg {r}_{\mathrm{mso}}/a$ ⁴ limit. In fact, this behavior could be interpreted as a consequence of the rotational effects of the attractor, which disappear in the Newtonian limit. The existence of such a ${C}^{-}\gt {C}^{+}$ couple is very constrained since the extension of the orbital difference, ${r}_{\min }^{-}-{r}_{\min }^{+}$, in the ${C}^{-}\gt {C}^{+}$ case is very limited and depends on ${r}_{{ \mathcal M }}^{\pm };$ such toroidal configurations are more likely to collide with subsequent possible merging of the tori.The case ${()}^{-}\lt {()}^{+}$ in Equation (19) is shown in Figure 9, left panel, and scheme III of Figure 4—see also Figure 11. The possible specific angular momentum of the ${C}^{-}$ configuration with ${{\ell }}_{-}\lt -{{\ell }}_{+}$ (for ${C}^{+}\lt {C}^{-}$ or ${C}^{-}\lt {C}^{+}$) depends on the unstable topology of the ${()}^{+}$ torus. The instability of the ${()}^{+}\lt {()}^{-}$ couple must take place on ${()}^{+}$. When ${{\ell }}_{+}\in {\boldsymbol{L}}1$, then there is the maximum possible separation between the centers of the couple. Therefore, it is necessary to consider the angular momentum ${{\ell }}_{-}({r}_{\min }^{+})$, which is the lower limit of the range in Equation (18), and the specific angular momentum ${{\ell }}_{-}(\bar{{\mathfrak{r}}})=-{{\ell }}_{+}({r}_{\min }^{+})$, which is the upper limit of this range. We can establish the upper bound by considering the topology of the ${()}^{+}$ configuration and the constraints on the ranges of the angular momentum. On the other hand, the lower bound satisfies the relation ${{\ell }}_{-}({r}_{\min }^{+})\,\lt {{\ell }}_{-}\lt {{\ell }}_{-}(\bar{{\mathfrak{r}}})=-{{\ell }}_{+}({r}_{\min }^{+})$—see Table 2 and Pugliese & Stuchlík (2016a). In fact, it is necessary to know the radius ${r}_{\min }^{+}$, for a fixed range of ℓ₊, and then establish the range of angular momentum ℓ_- of the corotating torus centered in ${r}_{\min }^{+}$—see also Pugliese & Stuchlík (2015). Then, we can combine the restrictions provided by the angular momentum range and those derived from the condition on the relation of the two angular momenta with the results of Table 2. Therefore, having a ${C}_{3}^{+}$ torus, the ${()}^{-}$ torus can then be in any topology; this is analogous, but with some restrictions, to ${C}_{2}^{+}$, mainly for ${C}_{3}^{-}$ and ${C}_{2}^{-}$ for slow attractors ($a\lt {a}_{{\aleph }_{0}}=0.390781M\,{{\ell }}_{\mathrm{mso}}^{+}=-{{\ell }}_{\mathrm{mbo}}^{-}$). For a ${C}_{1}^{+}$ torus, there is only ${C}^{-}\ne {C}_{3}^{-}$ for $a\lt {a}_{{\aleph }_{0}}$. If ${C}^{-}={C}_{1}^{-}$ or ${C}_{2}^{-}$, then ${C}^{+}$ can be in any angular momentum range, but subjected to several restrictions if orbiting slower attractors. However, these results have to be combined with those presented in Table 2. If ${C}^{-}={C}_{2}^{-}$, we have only the constraints provided by the complementary geodesic structure given in Table 2, while if ${C}^{-}={C}_{3}^{-}$ then, for $a\lt {a}_{{\aleph }_{0}}$, we have ${C}^{+}={C}_{2}^{+}$ or ${C}_{3}^{+}$.

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Table 2.  ℓcounterrotating Couples: Decoration of Bichromatic Vertices with Angular Momentum Classes ${\boldsymbol{Li}}$ of the Kerr Geometries with Dimensionless Spin $a\in ]0,M]$

Criticality:	Couples:		${()}^{+}\lt {()}^{-}$	Couples:		${()}^{-}\lt {()}^{+}$
${C}_{\times }^{-}\,\prec \,{{\rm{O}}}_{\times }^{+}\,:\,a\gt {a}_{{\gamma }_{+}}^{-}$	${({)}_{3}^{+}\lt ()}^{-}$	↦	$({)}_{3}^{+}\lt ({)}_{3}^{-}$	${({)}_{3}^{-}\lt ()}^{+}$	↦	$a\gt {a}_{{}_{u}}$ : $({)}_{3}^{-}\lt ({)}_{+}^{i}$
						$a\lt {a}_{{}_{u}}\,:\,({)}_{3}^{-}\lt ({)}_{3}^{+}$ $({)}_{3}^{-}\lt ({)}_{2}^{+}$
${{\rm{O}}}_{\times }^{-}\,\prec \,{{\rm{O}}}_{\times }^{+}\,:\,a\gt {a}_{\iota }$	${()}^{+}\lt ({)}_{3}^{-}$	↦	$({)}_{i}^{+}\lt ({)}_{3}^{-}$	${()}^{-}\lt ({)}_{3}^{+}$	↦	$({)}_{i}^{-}\lt ({)}_{3}^{+}$
${{\rm{O}}}_{\times }^{-}\,\prec \,{C}_{\times }^{+}$	${({)}_{2}^{+}\lt ()}^{-}$	↦	$a\gt {a}_{{}_{u}}\,:\,({)}_{2}^{+}\lt ({)}_{3}^{-}$	${({)}_{2}^{-}\lt ()}^{+}$	↦	$({)}_{2}^{-}\lt ({)}_{i}^{+}$
			$a\lt {a}_{{}_{u}}\,:\,({)}_{2}^{+}\lt ({)}_{3}^{-}$, $({)}_{2}^{+}\lt ({)}_{2}^{-}$
${C}_{\times }^{-}\,\prec \,{C}_{\times }^{+}$	${()}^{+}\lt ({)}_{2}^{-}$	$\mapsto$	$a\gt {\breve{a}}_{\aleph }\,:\,\nexists$	${()}^{-}\lt ({)}_{2}^{+}$	↦	$({)}_{i}^{-}\lt ({)}_{2}^{+}$
			$a\in ]{a}_{{}_{u}},{\breve{a}}_{\aleph }[\,:\,({)}_{1}^{+}\lt ({)}_{2}^{-}$
			$a\lt {a}_{{}_{u}}\,:\,({)}_{1}^{+}\lt ({)}_{2}^{-}$ $({)}_{2}^{+}\lt ({)}_{2}^{-}$
	${({)}_{1}^{+}\lt ()}^{-}$	↦	$a\gt {\breve{a}}_{\aleph }\,:\,({)}_{1}^{+}\lt ({)}_{3}^{-}$	${({)}_{1}^{-}\lt ()}^{+}$	↦	$({)}_{1}^{-}\lt ({)}_{i}^{+}$
			$a\in ]{\tilde{a}}_{\aleph },{\breve{a}}_{\aleph }[\,:\,({)}_{1}^{+}\lt ({)}_{2}^{-}$ $({)}_{1}^{+}\lt ({)}_{3}^{-}$
			$a\lt {\tilde{a}}_{\aleph }\,:\,$ $({)}_{1}^{+}\lt ({)}_{i}^{-}$
	${()}^{+}\lt ({)}_{1}^{-}$	↦	$a\gt {\tilde{a}}_{\aleph }\,:\,\nexists$	${()}^{-}\lt ({)}_{1}^{+}$	↦	$a\gt {a}_{{}_{u}}\,:\,({)}_{i}^{-}\lt ({)}_{1}^{+}$
			$a\lt {\tilde{a}}_{\aleph }:\,$ $({)}_{1}^{+}\lt ({)}_{1}^{-}$			$a\lt {a}_{{}_{u}}\,:\,({)}_{1}^{-}\lt ({)}_{1}^{+}\,({)}_{2}^{-}\lt ({)}_{1}^{+}$

Note.  State lines are in Figure 14, graphs are in Figure 9. Comments can be found in Appendix B—see also Figure 2. Definitions of spins are in Table 1.

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Sections 3.2.1 and 3.2.2 are dedicated to ℓcounterrotating couples, focusing on the sequentiality. To conclude this analysis, we summarize the situation in the following points:

• 1.  
  $$\begin{eqnarray}&&{\text{For}()}^{-}\lt {()}^{+}\quad {\rm{there}}\,{\rm{is}}\quad | {{\ell }}_{-}/{{\ell }}_{+}| \lt 1\\ &&\quad {\rm{and}}\,{\rm{}}\quad {{\ell }}_{-}\in ]{{\ell }}_{\mathrm{mso}}^{-},{{\ell }}_{-}({r}_{\min }^{+})[;\end{eqnarray} \tag{ 20 }$$

  see Section 3.2.2 and Equation (19). This case is detailed in Appendix B.0.2, where additional restrictions are discussed. The relation between the instability points (for ${\ell }\notin {\boldsymbol{L}}3$), which is the critical sequentiality, is fully addressed in Appendix B.0.3 and Table 2.In fact, the angular momenta of the tori in ${()}^{-}\lt {()}^{+}$ are fixed in the second column of Table 2, while the locations of the eventual instability points have been established in Figure 14 and the first column of Table 2. Since the distance between the radii ${r}_{\mathrm{mso}}^{+}\gt {r}_{\mathrm{mso}}^{-}$ increases with the increasing spin of the attractor, for some ranges of angular momenta, the critical points of the outer counterrotating configuration are ${r}_{\mathrm{mso}}^{-}\lt {r}_{\max }^{+}\lt {r}_{\mathrm{mso}}^{+}$, and the couple in Equation (20) are ${()}^{-}\lt {()}^{+}$, with ${()}^{-}\,\prec \,{()}^{+}$, as shown in Table 2. Therefore, this couple is a ${\bar{{\mathfrak{C}}}}_{{{\mathfrak{1}}}_{{\mathfrak{a}}}}$ one within the conditions of Equation (14) for ${r}_{\mathrm{mso}}^{-}\lt {r}_{\max }^{+}\lt {r}_{\min }^{-}\lt {r}_{\min }^{+}$, or a ${\bar{{\mathfrak{C}}}}_{{\mathfrak{0}}}$ of Equation (9) if ${r}_{\mathrm{mso}}^{-}\lt {r}_{\min }^{-}\lt {r}_{\max }^{+}\lt {r}_{\mathrm{mso}}\lt {r}_{\min }^{+}$. Therefore, the situation depends on the angular momentum of the outer configuration and the class of the attractor. On the other hand, if ${r}_{\max }^{+}\lt {r}_{\mathrm{mso}}^{-}$, then assuming ${\tilde{{\ell }}}_{-}\equiv {{\ell }}_{-}({r}_{\max }^{+})\in ]{{\ell }}_{\mathrm{mso}}^{-},{{\ell }}_{-}({r}_{\min }^{+})[$, this angular momentum differentiates between the configurations with ${{\ell }}_{-}\in ]{{\ell }}_{\mathrm{mso}}^{-},{{\ell }}_{-}({r}_{\max }^{+})[$, where ${r}_{\max }^{-}\gt {r}_{\max }^{+}$, implying ${()}^{-}\lt {()}^{+}$, with ${()}^{-}\succ {()}^{+}$, which is a ${\bar{{\mathfrak{C}}}}_{1b}$ class from Equation (15), from those with ${{\ell }}_{-}\in ]{{\ell }}_{-}({r}_{\max }^{+}),{{\ell }}_{-}({r}_{\min }^{+})[$, where ${r}_{\max }^{-}\lt {r}_{\max }^{+}$, implying ${()}^{-}\lt {()}^{+}$, with ${()}^{-}\,\prec \,{()}^{+}$, which is a ${\bar{{\mathfrak{C}}}}_{1a}$ class from Equation (12).
• 2.  
  Conversely, for the couple of tori

  $$\begin{eqnarray}&&\,{()}^{-}\gt {()}^{+}\quad {\rm{there}}\,{\rm{is}}\,{\rm{either}}\\ &&\quad {\bar{{\mathfrak{C}}}}_{1a}:\,{C}^{-}\gt {()}^{+}\quad {\rm{and}}\quad {C}^{-}\succ {()}^{+}\\ &&\quad {\rm{within}}\,{\rm{conditions}}\,{\rm{of}}\,{\rm{Equation}}\,{\rm{(14)}}\,{\rm{or}}\\ &&\quad {\bar{{\mathfrak{C}}}}_{1b}:\,{C}^{-}\gt {()}^{+}\quad {\rm{and}}\quad {C}^{-}\,\prec \,{()}^{+}.\\ &&\quad {\rm{There}}\,{\rm{can}}\,{\rm{be}}\quad | {{\ell }}_{-}/{{\ell }}_{+}| \gt 1\quad {\rm{if}}\quad {r}_{\min }^{-}\gt {\bar{{\mathfrak{r}}}}_{-},\\ &&\quad {\rm{then}}\,{\rm{we}}\,{\rm{obtain}}\,{\rm{}}\quad {\bar{{\mathfrak{C}}}}_{1b}\quad {\rm{according}}\,{\rm{to}}\,{\rm{Equation}}\,{\rm{(16),}}\,{\rm{}}\\ &&\quad {\rm{or}}\,{\rm{there}}\,{\rm{is}}\,{\rm{}}\quad | {{\ell }}_{-}/{{\ell }}_{+}| \lt 1\quad {\rm{if}}\quad {r}_{\min }^{-}\in ]{r}_{\min }^{+},{\bar{{\mathfrak{r}}}}_{-}[,\\ &&\quad {\rm{where}}\quad {\bar{{\mathfrak{r}}}}_{-}:\,{{\ell }}_{-}({\bar{{\mathfrak{r}}}}_{-})=-{{\ell }}_{+}.\end{eqnarray} \tag{ 21 }$$

The outer corotating torus of this couple cannot be unstable, as shown in Section 3.2.1.

Finally, we conclude this section by mentioning the couples with

$$\begin{eqnarray}&&{{\ell }}_{i/o}=-1\quad {\rm{which}}\,{\rm{corresponds}}\,{\rm{to}}\,{\rm{the}}\,{\rm{case}}\\ &&\quad {\bar{{\mathfrak{C}}}}_{{1}_{b}}\quad {\rm{considered}}\,{\rm{in}}\,{\rm{Equation}}\,{\rm{(15)}},\end{eqnarray} \tag{ 22 }$$

also discussed in Pugliese & Stuchlík (2015) as limiting cases for the perturbation analysis and as a limiting situation of $| {{\ell }}_{i/o}| \lessgtr 1$.

The evolution of these systems is fully described in the graphs of Figure 9. Comparing the graphs of Figures 8 and 9, it is clear that in the ℓcounterrotating case both vertices of a state may evolve. As a consequence of this, a change in the central state of the graph (which is also the initial state, the graph having only a subsequent section) generally heavily deforms the entire graph, as it is strongly dependent on the initial data (the decorations of the state vertices). The evolution of a state line is highly constrained by the initial decoration, as can be seen by comparing Figure 13 for the ℓcorotating states and Figure 14 and Table 2 for the ℓcounterrotating states. Consequently, we only have a limited number of possible states and evolutive lines for an ℓcounterrotating system: fixing the range of angular momenta for the initially separated couple (implying constraints on the K-parameters—see Section 3.3 and also Pugliese & Stuchlík 2015), we obtain rather stringent constraints from which it might be possible to predict to a large extent the existence and stability of the (isolated) couple of rotating tori around a spinning central BH.

For completeness, we also consider the configurations ${{\rm{O}}}_{\times }$ whose existence implies a relaxation of the condition of non-penetration of matter—we refer to Appendix A for further discussion. In Section 3.2.1, we focus on the ${()}^{+}\lt {()}^{-}$ double system introduced in Equation (21), while in Section 3.2.2 we investigate the ${()}^{-}\lt {()}^{+}$ couples introduced in Equation (20).

3.2.1. The ${\ell }{counterrotating}$ Configurations I: ${C}^{+}\lt {C}^{-}$

We start by exploring the bichromatic graph centered on the initial ${C}^{+}\lt {C}^{-}$ state in equilibrium, sketched in scheme III of Figure 4; examples of Boyer surfaces are in Figure 3. The second column of Figure 14 shows the set of possible states of these configurations, and details on the sequentiality are provided in Table 2. We discussed the configuration sequentiality following Equation (20). The graph in Figure 9, left panel, describes all the possible evolutive phases of the centered ${()}^{+}\lt {()}^{-}$ system. We can therefore derive some conclusions by comparing with the graph of Figure 8 for the ℓcorotating couples, which also describes a bichromatic graph in a static (a = 0) spacetime. Similarly to the ℓcorotating case, the state lines and their evolution are essentially independent from the class of the attractors.

Considering the cases where the equilibrium state may evolve toward the ${C}_{\times }$ topologies associated with accretion, we conclude that if the inner torus is accreting, then, similarly to the ℓcorotating torus and to the bichromatic graph in the static geometry, the system can evolve only into a state where the outer torus is in its equilibrium topology (the vertex ${C}_{\times }^{+}$ is connected to only one state line). Moreover, as collision between two tori in their equilibrium states is in general possible, any instability of the outer torus is inevitably preceded by the destruction of the macro-configuration. In fact, an inversion in the critical sequentiality is not possible for this couple. The case of a bichromatic graph with the central state ${C}^{+}\lt {C}^{-}$ is indeed similar to the bichromatic graph representing an ℓcorotating couple (or the case of static spacetime): mono- or bichromatic graphs in static spacetime and bichromatic ones where ${C}^{+}\lt {C}^{-}$ for a Kerr geometry are indistinguishable on many aspects based on the states' properties and evolutions. In the investigation of the collision in the bichromatic graph at $a\ne 0$, we should consider the opposite relative rotation of the tori. Finally, we note that since the inner torus is counterrotating with respect to the attractor, this system will be confined to the orbital range $r\gt {r}_{\mathrm{mbo}}^{+}$, because for some topologies, as is clear from Pugliese & Stuchlík (2016a), the inner margin of the torus may be in $]{r}_{\mathrm{mbo}},{r}_{\mathrm{mso}}]$, while the tori must be centered at $r\gt {r}_{\mathrm{mso}}^{+}$.

If ${\ell }\in {\boldsymbol{L}}1$ or ${\boldsymbol{L}}2$, all of these configurations are described by Equation (21), and therefore they cannot constitute a ${{\mathfrak{C}}}_{0}$ system. It is therefore evident from Equation (21), and also from the peculiar sequentiality of the couples, that ${{\mathfrak{C}}}_{0}$ configurations show strong similarities with the couple described by the monochromatic graphs. Furthermore, from Table 2 and considering also Equation (21), we find that the ${\bar{{\mathfrak{C}}}}_{{1}_{b}}$–${C}^{+}\lt {C}^{-}$ couples are

$$\begin{eqnarray}&&{\bar{{\mathfrak{C}}}}_{{1}_{b}}\,:\quad ({)}_{1}^{+}\lt ({)}_{1}^{-}\quad \forall a,\quad {\rm{and}}\\ &&\quad ({)}_{2}^{+}\lt ({)}_{2}^{-}\quad {\rm{for}}\quad a\in ]{a}_{\iota },{a}_{{}_{u}}[.\end{eqnarray} \tag{ 23 }$$

However, a vertex could also be a ${C}_{3}$ configuration, and it may be associated with the first phases of torus formation, being far enough from the attractor (${r}_{\min }\gt {\bar{{\mathfrak{r}}}}_{\gamma }$) and with a large specific angular momentum (${\ell }\gt {{\ell }}_{\gamma }$). The magnitude of the specific angular momentum of the torus would, during its evolution, decrease. In this last case, a decrease of the specific angular momentum magnitude from a ${C}_{3}$ configuration could be preceded by an ${{\rm{O}}}_{\times }$ topology, since the specific angular momentum transition would be ${\boldsymbol{L}}3$ to ${\boldsymbol{L}}1$ through ${\boldsymbol{L}}2$. We see that this configuration should be the most difficult to observe because its formation is strongly constrained by the attractor spin. More generally, from the second column of Table 2, we can draw the following evolutionary schemes (a further discussion regarding loops for these couples is in Appendix A.1)

• 1.  
  Accretion: ${C}_{\times }^{+}\lt {C}^{-}$ and the final states of evolution. The macro-configuration with state ${C}_{\times }^{+}\lt {C}^{-}$ must be a ${\bar{{\mathfrak{C}}}}_{1b}$ one, unless the outer corotating disk is in ${C}_{3}^{-}$, which is only possible for the attractors with $a\lt {\tilde{a}}_{\aleph }$ (this can be seen by considering the first and second columns of Table 2 and the results of Equation (21)). Therefore, the couples of ${\bar{{\mathfrak{C}}}}_{1a}$ cannot lead to accretion, and any instability in one torus of the couple will destroy the couple. Then, in the fields of faster attractors, the specific angular momentum of the outer disk cannot decrease to ${\boldsymbol{L}}1$ without destruction of the macro-configuration. Consequently, we arrive at the remarkable conclusion that for a slow attractor with $a\lt {\tilde{a}}_{\aleph }$, there must be $({)}_{1}^{+}\lt ({)}_{1}^{-}$—see Figure 3.This means that such a double system is possible exclusively in the geometry of slow rotating attractors, with tori centered in $]{r}_{\mathrm{mso}}^{+},{\bar{{\mathfrak{r}}}}_{\mathrm{mbo}}^{+}[$ and $]{r}_{\mathrm{mso}}^{-},{\bar{{\mathfrak{r}}}}_{\mathrm{mbo}}^{-}[$, respectively. The second notable result is that such a couple is the only couple possible with ${()}^{+}\lt ({)}_{1}^{-}$ and must exist in the fields of these slow attractors. Assuming that the inner torus has been formed before or simultaneously with the formation of the outer torus, the final states⁵ $({)}_{1}^{+}\lt ({)}_{1}^{-}$ of their evolution can be reached only around attractors with $a\lt {\tilde{a}}_{\aleph }$, when the outer torus reaches the angular momentum ${\boldsymbol{L}}1$. Thus, the outer corotating torus may be in its last stage of evolution only if the inner counterrotating one is $({)}_{1}^{-}$; otherwise, the ringed accretion disk would be destroyed due to the merging of the two tori. Any instability in the outer torus would in any case lead to the destruction of the macro-configuration, which therefore seems to be unlikely to exist in the "old" systems where the tori have reached their last evolutive stages, and they should be a feature of relatively "young" systems. This can be seen as a strong indication that the ${()}^{+}\lt {()}^{-}$ couples may not be frequent, with the exclusion of the recent population of Kerr BH attractors.
• 2.  
  Accretion: ${C}_{\times }^{+}\lt {C}^{-}$ and the initial states of evolution toward accretion. If the attractor is slow enough, i.e., $a\lt {\tilde{a}}_{\aleph }$, then the outer torus ${C}_{o}^{-}$ can be anywhere according to the range of specific angular momenta, since it is part of a system where the inner counterrotating torus is $({)}_{1}^{+}$. This means that the formation of such a double system is most likely in those geometries. On the other hand, if the tori orbit a fast attractor with $a\gt {\breve{a}}_{\aleph }$, then the couple can form only during the earliest stages when the corotating torus has a large angular momentum, i.e., ${C}_{+}^{-}={()}_{3}$ for $a\gt {\breve{a}}_{\aleph }$, or ${C}_{o}^{-}={()}_{3}$ or ${C}_{o}^{-}={()}_{2}$ for $]{\tilde{a}}_{\aleph },{\breve{a}}_{\aleph }[$—see details in Table 2.
• 3.  
  Formation of the couple and the early stages of evolution. During the evolution from an equilibrium torus $C$ to an unstable (accretion) topology ${C}_{\times }^{1}$, the magnitude of the specific angular momentum of the torus generally decreases, preserving the state sequentiality. Then we can provide constraints on the formation of these couples by identifying the conditions for the appearance of these couples, which form in some geometries at some stages in the evolution of the inner counterrotating torus toward accretion. To carry out these arguments, we assume three hypothetical stages of the torus evolution: an early stage formed as a ${C}_{3}^{+}$, an intermediate ${C}_{2}^{+}$ one, and the final ${C}_{1}^{+}$ stage eventually leading to ${C}_{\times }^{+}$. On the other hand, a torus may be formed in any of these stages. We prove that these couples may be formed only in certain stages of the inner torus evolution for some Kerr attractors. This analysis in turn sets significant limits on the observational investigation of these systems, providing constraints on the tori–attractor system, and it is able to impose some constraints on the central attractor of an observed couple.From Table 2, we see that configurations formed very far from the attractor and with a large angular momentum magnitude are strongly constrained. If the inner torus is formed as a ${C}_{i}^{+}={C}_{3}^{+}$ one, then at this stage the outer torus must  necessarily be a ${C}_{3}^{-}$ one with a large angular momentum magnitude; any other solution would inevitably lead to the collision of the two tori. This means that the possible formation of a second corotating torus in the early stages of the formation of the counterrotating one is severely limited. Conversely, it is clear that a torus with a large angular momentum may be formed under any circumstances not undermining the evolution of the first vertex of the state and therefore its evolutive line.A more complicated situation occurs if the inner torus is in its intermediate stage with ${\ell }={{\ell }}_{2}$. The outer torus must then in all cases have stringent conditions on its specific angular momentum, and the situation also depends on the attractor spin: if $a\gt {a}_{\iota }$, then only an outer ${C}_{3}^{-}$ torus may be formed, thus reducing the possibility of the formation of the double torus around the fastest attractors. In the geometry of slower attractors where $a\lt {a}_{\iota }$, the outer torus may be in ${()}_{2}$.When the outer corotating torus is ${C}_{2}^{-}$, double-torus systems cannot orbit faster attractors with $a\gt {\breve{a}}_{\aleph }$, while for dimensionless spin $a\in ]{a}_{{}_{u}},{\breve{a}}_{\aleph }[$, the inner torus must be in $({)}_{1}^{+}$. For slower attractors with $a\lt {a}_{{}_{u}}$, the inner counterrotating torus must be a $({)}_{1}^{+}$ or a $({)}_{2}^{+}$ one.

3.2.2. The ${\ell }{counterrotating}$ Couple II: ${C}^{-}\lt {C}^{+}$

This section is focused on the ℓcounterrotating configurations with ${()}^{-}\lt {()}^{+}$, sketched in scheme IV of Figure 4.

The bichromatic graph, centered on the initial equilibrium state ${C}^{-}\lt {C}^{+}$, is in Figure 9, right panel. The possible states are listed in Figure 14, and details on the sequentiality can be found in Table 2.

This case significantly differs from the ${()}^{+}\lt {()}^{-}$ one, as illustrated by the graph of Figure 9, right panel, and discussed in Section 3.2.1. The major difference for a ${()}^{-}\lt {()}^{+}$ system is due to the distinctive double geodesic structure of the Kerr spacetime in the case of a corotating inner torus, where in fact the critical sequentiality is not uniquely determined by the configuration sequentiality. By comparing the two graphs of Figure 9, we note that for the ${()}^{-}\lt {()}^{+}$ state, there are more state lines connected by the evolutive lines for the inner vertex than the outer vertex—see also Figure 11. This means that the evolution toward instability may also occur for the ${\bar{{\mathfrak{C}}}}_{0}$ system from the second counterrotating vertex or even from both vertices: this variety of solutions makes this case less restrictive than the ${()}^{+}\lt {()}^{-}$ one, allowing different evolutive paths and favoring several possibilities for the formation of the torus couple.

On the other hand, from the third column of Table 2, which provides the necessary conditions for the fixed configuration sequentiality, we note that the states ${()}^{-}\lt {()}^{+}$ are distinguishable for different Kerr attractors only in the case of a ${C}_{3}^{-}\lt ({)}_{1}^{+}$ couple, which may be formed only in the geometries of fast Kerr attractors with $a\gt {a}_{{}_{u}}$. As we shall discuss at the end of this section, this has an important consequence on the formation of the couple and the eventual evolution toward accretion. This implies that in these geometries, in the early stages of formation of the corotating inner torus, an outer counterrotating torus can also be formed, evolving finally into a $({)}_{1}^{+}$ topology and leading eventually to accretion—Figure 11. Furthermore, as mentioned at the beginning of Section 3.2, the couples ${()}^{-}\lt {()}^{+}$ may give rise to a class transition from a $\bar{{{\mathfrak{C}}}_{{1}_{{\rm{a}}}}}$ or $\bar{{{\mathfrak{C}}}_{{1}_{{\rm{b}}}}}$ class (where instability of the outer torus is forbidden) to a $\bar{{{\mathfrak{C}}}_{0}}$ class, with a consequent change in the critical sequentiality—see Equations (9), (12), and (15). Such a transition implies that the final state fulfills the condition in Equation (10) for the specific angular momenta of the two tori.

On the other hand, we should consider the arrangement of the angular momenta as given in Figure 10 and the decoration of the initial state neglecting the size of the torus (the K parameter). More specifically, the class of the specific angular momentum for these torus couple configurations depends on the class of attractors and the constraints of Equation (10) for a ${\bar{{\mathfrak{C}}}}_{0}$ class. Then we need to consider Kerr attractors where both conditions on the specific angular momentum of the torus and the definitive constraints on the radii ${{\mathfrak{C}}}_{{1}_{{\rm{a}}}}$ or ${{\mathfrak{C}}}_{{1}_{{\rm{b}}}}$ and the final state ${{\mathfrak{C}}}_{0}$ given by Equations (12) or (15), respectively, and the last one of Equation (9) are met—see also Figures 1, 2, 10.

Assuming the transition , in accordance with Equations (12) and (13), the state in ${{\mathfrak{C}}}_{{1}_{{\rm{a}}}}$ must necessarily be in equilibrium or a ${C}^{-}\lt {C}^{+}$ couple, which means that if the inner torus is accreting onto the attractor, it cannot lead to a class transition.

Considering separately the possibility of ${C}_{3}$ states, we focus on the toroidal configurations covered by the classification in Equations (9), (12), and (15), and we list here the states ${()}^{-}\lt {()}^{+}$ in these different classes. From Table 2, and considering also Equations (9) and (12), we obtain

$$\begin{eqnarray}&&{\bar{{\mathfrak{C}}}}_{1b}\,:\quad ({)}_{1}^{-}\lt {C}_{2}^{+}\quad {\rm{for}}\quad a\lt {a}_{{\gamma }_{+}}^{-}\\ &&\quad {\rm{or}}\quad ({)}_{2}^{-}\lt {C}_{2}^{+}\quad {\rm{for}}\quad a\lt {a}_{\iota },\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&\bar{{{\mathfrak{C}}}_{{\mathfrak{0}}}}\,:\quad ({)}_{1}^{-}\lt ({)}_{1}^{+},\quad ({)}_{2}^{-}\lt ({)}_{1}^{+}\quad \quad {\rm{for}}\quad a\gt {a}_{\iota },\\ &&\quad ({)}_{1}^{-}\lt ({)}_{2}^{+}\quad {\rm{for}}\quad a\gt {a}_{{\gamma }_{+}}^{-},\,{\rm{according}}\,{\rm{to}}\,{\rm{Equation}}\ (10),\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&{\bar{{\mathfrak{C}}}}_{1a}\,:\quad ({)}_{1}^{-}\lt {C}_{1}^{+},\quad ({)}_{2}^{-}\lt {C}_{1}^{+},\,({)}_{2}^{-}\lt {C}_{2}^{+}\\ &&\quad {\rm{for}}\quad a\gt {a}_{\iota },\quad ({)}_{1}^{-}\lt {C}_{2}^{+}\quad {\rm{for}}\quad a\gt {a}_{{\gamma }_{+}}^{-}.\end{eqnarray} \tag{ 26 }$$

where we used the property of $\bar{{{\mathfrak{C}}}_{{1}_{b}}}$ for Equation (24), property Equation (9) for Equation (25), and property Equation (12) for Equation (26). Finally in Equation (15), the first and third columns of Table 2 has been taken into account.

In the following, we will concentrate primarily on the $C$ and ${C}_{\times }$ topologies, referring to Equation (20) for the relation between specific angular momentum and keeping in mind the results of Table 2. Further discussions regarding loops in these ℓcounterrotating couples are in Appendix A.1.

• 1.  
  Accretion and the final states of evolution. The following accretion states are possible:

  $$\begin{eqnarray}&&\,{C}^{-}\lt \prec \,{C}_{\times }^{+}\quad (\bar{{{\mathfrak{C}}}_{{\mathfrak{0}}}}),\quad {C}_{\times }^{-}\lt \,\prec {C}_{\times }^{+}\quad (\bar{{{\mathfrak{C}}}_{{\mathfrak{0}}}});\\ &&\quad {C}_{\times }^{-}\lt {C}^{+}\quad {\rm{with}}\,{\rm{an}}\,{\rm{antecedent}}\,{\rm{state}}\quad {C}^{-}\lt {C}^{+};\end{eqnarray} \tag{ 27 }$$

  see Figure 9. Geometrical correlation, and then collision, is generally possible. The critical sequentiality of the couple remains undetermined if the outer vertex is in equilibrium—see Table 2. If the outer vertex is unstable, then it must be a ${\bar{{\mathfrak{C}}}}_{0}$ class of Equation (25) (for ${{\ell }}_{-}\in {\boldsymbol{L}}1$ or ${\boldsymbol{L}}2$). If the outer torus is in equilibrium, then it may be a ${\bar{{\mathfrak{C}}}}_{{1}_{b}}$, ${\bar{{\mathfrak{C}}}}_{{1}_{a}}$, or also ${\bar{{\mathfrak{C}}}}_{0}$ class, according to Equations (25) and (26). Considering the third column of Table 2, if the inner torus is in the final stage of evolution, eventually accreting onto the BH, then the outer torus could acquire any angular momentum.⁶ However,  if the outer counterrotating torus is $({)}_{1}^{+}$ and it is in its last evolutive phase according to the evolutive framework assumed here, then the inner corotating ring could be in any evolutive stage (as long as the constraint of no penetration of matter is fulfilled) if they are orbiting fast attractors with $a\gt {a}_{{}_{u}}$. The formation of a $({)}_{1}^{+}$ outer torus is in principle possible at any stage of evolution of the inner torus (i.e., for any ℓ_i). On the other hand, for slow attractors with $a\lt {a}_{{}_{u}}$, the corotating ring must be in an intermediate phase or in its last evolutive phase. As mentioned earlier, the existence of a couple ${C}_{3}^{-}\lt ({)}_{1}^{+}$ is possible only for Kerr attractors with $a\gt {a}_{{}_{u}}$—see Table 2.Finally, accretion from the outer configuration may be possible only in class ${\bar{{\mathfrak{C}}}}_{0}$ of Equation (9), and in accordance with the constraints of Equation (10), could also be a consequence of the transition from an equilibrium state in ${\bar{{\mathfrak{C}}}}_{{1}_{a}}$ or ${\bar{{\mathfrak{C}}}}_{{1}_{b}}$.We focus on the emergence of an unstable phase for the outer vertex corresponding to the last configuration to be formed.⁷ Remarkably, the outer configuration can be accreting for each attractor, but for slow attractors it is limited only to the final stages of evolutions ${()}_{2}$ and ${()}_{1}$, for the corotating inner ring, which cannot be ${C}_{3}^{-}$. For slow attractors $a\lt {a}_{{}_{u}}$, the outer torus cannot accrete from an inner corotating torus in the early stages of development, ${C}_{3}^{-}$; this is prohibited due to Table 2. We achieve the remarkable result that for an accreting torus corotating with the Kerr attractor, there is no inner corotating or counterrotating torus between the accreting torus and the attractor. On the other hand, there can only be an inner corotating torus if the outer accreting ring is counterrotating with respect to the attractor (an outer torus in accretion is also forbidden in the couples formed by the ℓcorotating surfaces with counterrotating tori). Finally, we note that the class of the angular momentum of the inner torus can be inferred from the results of Table 2.
• 2.  
  Accretion: intermediate phases. ${()}_{2}$ We now focus on the intermediate ${()}_{2}$ evolutive phases. The considerations outlined in Equation (24) hold. We note that this phase is the one requiring in general fewer constraints on the vertex decoration. In fact, for both $({)}_{2}^{\pm }$ cases, each of the two vertices may be, independently of the spin attractor, in any evolutive stage, considered to be the other in ${()}_{2}$. For all these reasons, we can say that the formation and stability of such a couple with a ${()}_{2}$ configuration is less constrained, while the formation and stability of a ${()}^{-}\lt {()}^{+}$ couple would be hampered in the earliest or latest evolutive stages.
• 3.  
  Couple formation and the early stages of evolution. We now consider the case of toroidal configurations that are far away from  the attractor ($r\gt {r}_{\gamma }^{\pm }$) in the first phase of their formation and with large specific angular momentum magnitude ($\mp {{\ell }}_{\pm }\gt \mp {{\ell }}_{\gamma }^{\pm }$). First, from Table 2 we see that the inner vertex can be in any topology if the outer configuration is ${C}_{3}^{+}$, which is associated with the earliest stages of formation.If the inner ring is ${C}_{3}^{-}$, then for large spin, $a\gt {a}_{{}_{u}}$, a double system may be formed with the outer torus allowed to have a different angular momentum (see also the case of ${C}^{+}\lt {C}^{-}$ in the second column of Table 2), whereas for slow attractors, $a\lt {a}_{{}_{u}}$, only configurations $({)}_{3}^{+}$ and $({)}_{2}^{+}$ could be considered to form a double system and therefore to be initial states toward accretion. This may be important when the formation of the double system occurs almost simultaneously, or if the double tori can be formed in the later phases of the life of the inner torus–attractor system. In this evolutive scheme, we could say that these tori can also be formed almost simultaneously in any Kerr geometry, but in the spacetimes of the slow attractors, the outer torus must have a sufficiently large specific angular momentum. Thus, these couples are probably formed around faster attractors.
• 4.  
  Evolution paths toward accretion. The evolution of the $C$ topology toward the accretion phase ${C}_{\times }$ might generally happen along several different paths according to the initial specific angular momentum of the equilibrium configuration. In fact, it could possibly give rise to a composite evolutive line, involving more than two state lines and determined by the composition of two intermediate states in which, for example from an initial ${C}_{3}$ configuration, the torus reaches, due to loss of specific angular momentum magnitude, the ${C}_{\times }^{1}$ topology of accretion. Then, referring to Figure 9, right panel, and avoiding the discussion of possible loops, we concentrate on the part of the graph formed only by the vertices $({C}_{\pm },{C}_{\times }^{\pm })$. Loops are discussed in Appendix A.1. We suppose that the accreting inner torus, reaching its maximum elongation on the equatorial plane $\lambda ={\lambda }_{\times }$, does not collide with the outer vertex (the related conditions are addressed in Section 3.3). Then we obtain two possible processes:

  

  demonstrated in Figure 9. Process $({\rm{a}})$ of Equation (28) may not involve an evolution of the outer configuration, which remains, in accordance with the constraints discussed in Table 2, in the equilibrium topology. On the other hand, for a state in ${\bar{{\mathfrak{C}}}}_{0}$, the outer torus can reach the stage of accretion prior to or together with the inner torus, according to the evolutive lines of the graph in Figure 9. In this case, by considering also Equation (24), we obtain the following two evolutive paths:

  

  Path $({\rm{b}})$ of Equation (28) and path $({\rm{d}})$ of Equation (29) represent an extension of paths $({\rm{a}})$ and $({\rm{c}})$, respectively. Assuming that after (or simultaneously with) the emergence of the instability of one vertex, an instability also in the other vertex of state may occur, which may be independent from the other instability. This is contrary to situations as with the ${()}^{+}\lt {()}^{-}$ couples, where such an extension is not possible because it must be preceded by merging with destruction of the couple.⁸

3.3. Collisions, Emergence of the ${{\boldsymbol{C}}}_{\odot }^{2}$ Macro-Configuration, and Merging

Collision between the tori may take place as a consequence of the following mechanisms: in the couple ${()}^{-}\lt {C}_{1}^{+}$, it may occur only as the impact of the inner Roche lobe of the outer accreting torus on the inner torus. Conversely, collision may involve only an evolution of the outer Roche lobe of the inner torus, even for the two equilibrium tori, with the formation of a ${{\boldsymbol{C}}}_{\odot }^{2}$ or ${C}_{\odot }^{x}{}^{2}$ macro-configuration where ${y}_{1}^{i}={y}_{3}^{o}$. The effective potential for such a ringed disk is

$$\begin{eqnarray}&&{{V}_{\mathrm{eff}}^{{C}_{\odot }^{2}}| }_{{K}_{i}}={V}_{\mathrm{eff}}^{i}({{\ell }}_{i})\,{\rm{\Theta }}\,({y}_{\odot }-y)\,\bigcup \,{V}_{\mathrm{eff}}^{o}({{\ell }}_{o})\,{\rm{\Theta }}\,(y-{y}_{\odot }),\\ &&\quad {\rm{where}}\quad {y}_{\odot }\equiv {y}_{3}^{o}={y}_{1}^{i},\quad {\bar{\lambda }}_{{{\boldsymbol{C}}}_{\odot }}=0,\quad {\lambda }_{{{\boldsymbol{C}}}_{\odot }}={\lambda }_{i}+{\lambda }_{o};\end{eqnarray} \tag{ 30 }$$

see Figure 7. Such a system may arise as a consequence of the accretion of the inner ring, which, reaching its maximum elongation ${\lambda }_{\times }$ on the equatorial plane at the emergence of the instability, impacts on the outer equilibrium torus. It is necessary therefore to change the inner torus parameters only. On the other hand, it is clear that the condition ${y}_{1}^{i}={y}_{3}^{o}$ could follow also from a change in the outer torus morphology only, not involving an instability in any ring of the couple. More generally, for the occurrence of such collisions between two tori where the outer tori is quiescent, the conditions for correlation must be matched (Pugliese & Stuchlík 2016a). From Figures 14 and 13, we can infer the necessary conditions for the states of the ℓcounterrotating and ℓcorotating couples to be separated (not correlated), thus preventing collision. However, for all other states, we look for relations between the specific angular momenta $({{\ell }}_{i},{{\ell }}_{o})$ of the two tori such that it is possible to find a couple $({K}_{i},{K}_{o})$ for which a geometrical correlation can occur, implying the condition ${y}_{1}^{i}={y}_{3}^{o}$ in terms of the relations between the couples of parameters $({K}_{i},{K}_{o})$ and $({{\ell }}_{i},{{\ell }}_{o})$. First, the necessary conditions on the outer torus for collisions in the macro-configuration to occur are

$$\begin{eqnarray}&&\begin{array}{l}{\rm{for}}\ {()}_{o}\ne ({)}_{1}^{o}\,{\rm{\text{there has to be a well defined effective potential}}}\\ \quad {V}_{\mathrm{eff}}({{\ell }}_{o})\lt 1\quad {\rm{in}}\quad [{y}_{1}^{i},{r}_{\min }^{o}[,\end{array}\end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray}&&\begin{array}{l}\,{\rm{which}}\,{\rm{particularly}}\,{\rm{means:}}\\ {\rm{for}}\,{\rm{}}\quad {()}_{o}=({)}_{2}^{o}\quad {\rm{there}}\,{\rm{has}}\,{\rm{to}}\,{\rm{be}}\,{\rm{}}{y}_{1}^{i}\in ]\bar{{\mathfrak{r}}},{r}_{\min }^{o}\\ [{\rm{where}}\quad \bar{{\mathfrak{r}}}\in [{r}_{\mathrm{mbo}}^{o},{r}_{\min }^{o}[:\,{V}_{\mathrm{eff}}({{\ell }}_{o},\bar{{\mathfrak{r}}})=1\\ \end{array}\end{eqnarray} \tag{ 32 }$$

$$\begin{eqnarray}&&{\rm{while}}\,{\rm{for}}\quad {()}_{o}=({)}_{o}^{1}\quad {\rm{this}}\,{\rm{condition}}\,{\rm{has}}\,{\rm{to}}\,{\rm{be}}\,{\rm{supplied}}\,{\rm{with}}\,{\rm{}}\\ &&\quad {y}_{1}^{i}\in ]{r}_{\max }^{o},{r}_{\min }^{o}].\end{eqnarray} \tag{ 33 }$$

In general, these conditions hold for ${r}_{\min }^{i}\lt {r}_{\max }^{o}$ or ${r}_{\max }^{i}\lt {r}_{\max }^{o}\lt {r}_{\min }^{i}$ or also for ${r}_{\max }^{o}\lt {r}_{\max }^{i}\lt {r}_{\min }^{i}$. However, Equations (31)–(33) imply⁹ (1) ${r}_{\mathrm{mbo}}^{o}\in {C}_{i}$ or (2) ${r}_{\mathrm{mbo}}^{o}\lt {y}_{3}^{i}$. The first condition holds only for the ℓcounterrotating couples as described in Pugliese & Stuchlík (2016a); the second condition, however, also includes the ℓcorotating couples and also considers the case ${r}_{\mathrm{mbo}}^{o}\notin {C}_{i}$ which, for example, is always verified for ${C}^{+}\lt {C}^{-}$ where ${r}_{\mathrm{mbo}}^{o}\lt {r}_{\mathrm{mbo}}^{i}$.

The necessary conditions of the inner torus for collision to occur are:

$$\begin{eqnarray}&&{\rm{for}}\quad ({)}_{1}^{i}\,:\,{V}_{\mathrm{eff}}({{\ell }}_{i},{y}_{3}^{o})\leqslant {K}_{\max }^{i}\,,\end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray}&&{\rm{for}}\,({)}_{2}^{i}\,\nexists \,{\rm{always}}\,{\rm{}}{K}_{i}\,:\,(34)\ \text{is satisfied and, particularly the,}\\ &&\quad \text{following condition also always holds}\end{eqnarray} \tag{ 35 }$$

$$\begin{eqnarray}&&\begin{array}{l}{\rm{for}}\quad {C}_{3}^{i}:\,\nexists \,\bar{{\mathfrak{r}}}\lt {r}_{\min }^{i}:{V}_{\mathrm{eff}}(\bar{{\mathfrak{r}}},{{\ell }}_{i})={V}_{\mathrm{eff}}({y}_{3}^{o},{{\ell }}_{i})\\ \quad \text{and a well defined effective potential}\\ \qquad \,{V}_{\mathrm{eff}}({{\ell }}_{i})\lt 1\,{\rm{in}}\,[\bar{{\mathfrak{r}}},{r}_{\min }^{i}].\\ \end{array}\end{eqnarray} \tag{ 36 }$$

Note that there is ${r}_{\mathrm{mbo}}^{\pm }\notin {C}_{\pm }$ (Pugliese & Stuchlík 2016a). Condition (35) implies that for a ${C}_{2}^{i}$ torus it is always possible to find a proper K_i parameter such that ${y}_{1}^{i}={y}_{3}^{o}$—see also Table 2. Since for a ${C}_{2}^{o}$ torus there is $\sup {K}_{o}=1$, the condition ensuring that collision does not occur reads ${y}_{1}^{i}\lt {r}_{\sup }^{o}$, where ${r}_{\sup }^{o}:\,{V}_{\mathrm{eff}}^{o}({r}_{\sup }^{o})=1$, or the potential is not well-defined. This last condition also holds for a ${C}_{3}^{o}$ ring. However, for the characterization of collision in a ${{\boldsymbol{C}}}_{\odot }^{2}$ macro-configuration, we should simultaneously consider the conditions on the outer edge of the inner ring and on the morphology of the outer torus. For a $({)}_{1}^{i}$ ring, because collision does not occur, it has to be ${K}_{o}\,:\,{K}_{\max }^{i}\lt {V}_{\mathrm{eff}}^{i}({y}_{3}^{o})\lt 1$. Besides, a parameter ${K}_{i}\lt 1\,:\,{y}_{1}^{i}={y}_{3}^{o}$ can always exist for a ${C}_{2}^{i}$ disk, since ${K}_{\max }^{i}\geqslant 1$, with the effective potential well-defined for $r\gt {r}_{\min }$ and asymptotically ${V}_{\mathrm{eff}}=1$. These necessary but not sufficient conditions for collision imply a precise relation on the ring sequentiality, according to the constraints provided by Table 2. Further restrictions can be found by comparing the inclusion relations of the notable radii addressed in Pugliese & Stuchlík (2016a). As the ℓcorotating couples always form a ${\bar{{\mathfrak{C}}}}_{{1}_{b}}$ ringed disk, these configurations are more likely to lead to collision, particularly for the couples made by two ${C}_{1}^{\pm }$ tori, where collision is always possible. Other cases such as the ${C}_{1}^{\pm }\lt {C}_{2}^{\pm }$ couples imply satisfaction of the property ${V}_{\mathrm{eff}}({y}_{3}^{1},{{\ell }}_{2})\lt 1$, which is favored in the case where ${{\ell }}_{2}/{{\ell }}_{1}\approx 1$ and ${K}_{1}/{K}_{2}\gg 1$, or as $| {{\ell }}_{2}| =| {{\ell }}_{\mathrm{mbo}}| +{\epsilon }_{+}$ and $| {{\ell }}_{1}| =| {{\ell }}_{\mathrm{mbo}}| -{\epsilon }_{-}$, where ${\epsilon }_{\pm }\gtrapprox 0$. Analogous relations hold for ${C}_{2}^{\pm }\lt {C}_{2}^{\pm }$, ${C}_{2}^{\pm }\lt {C}_{3}^{\pm }$, and ${C}_{3}^{\pm }\lt {C}_{3}^{\pm }$. Furthermore, from analysis of critical and configuration sequentiality, Table 2 shows some necessary but not sufficient conditions for collision emergence, constraining also the ℓcorotating configurations, which, according to the only constraints of Equation (30), in principle might lead to collision. However, by considering the effective potential in Equation (8), we can obtain an immediate relation for colliding configurations in ${{\boldsymbol{C}}}_{\odot }^{2}$ in the ℓcorotating case:

$$\begin{eqnarray}&&{K}_{i}\in ]{K}_{\min }^{i},{V}_{\mathrm{eff}}({{\ell }}_{i},{r}_{\min }^{o})]\quad {\rm{and}}\\ &&\quad {K}_{o}\in ]{K}_{\min }^{o},{V}_{\mathrm{eff}}({{\ell }}_{o},{y}_{1}^{i})]\quad {\rm{for}}\,{\rm{}}\quad {{\ell }}_{i}{{\ell }}_{o}\gt 0\end{eqnarray} \tag{ 37 }$$

(Pugliese & Stuchlík 2015). Then Equation (37) is a necessary condition for a ringed disk of order two, represented in schemes I and II of Figure 4, to evolve into a ${{\boldsymbol{C}}}_{\odot }^{2}$ configuration.

On the other hand, the situation for an ℓcounterrotating couple is particularly complex, depending on state correlation and the possible sequentiality as outlined in Table 2. Moreover, for these couples, we cannot easily write down a condition analogous to Equation (37). This is due to the fact that, as seen in Section 3.2, for ℓcounterrotating couples, the order relation between the magnitude of the specific angular momenta and the location of the disks, and the effective potential at the minimum points is not straightforwardly traced. In the following we shall focus mainly on ℓcorotating couples.

Collision after growth of the outerring.We focus first on ℓcorotating couples. From Figure 14 we know that ${C}_{\pm }^{i}\lt {C}_{\pm }^{o}$ implies ${C}_{\pm }^{i}\succ {C}_{\pm }^{o}$, then we obtain that for ${{\ell }}_{i}\lt {{\ell }}_{o}$, when ${{\ell }}_{i}$ and ℓ_o are in ${\boldsymbol{L}}1$ or ${\boldsymbol{L}}2$, it is possible to find a proper K_o for the emergence of a ${{\boldsymbol{C}}}_{\odot }^{2}$ configuration. Therefore, for an initially separated couple with $\{{{\ell }}_{i},{{\ell }}_{o}\}$ in ${\boldsymbol{L}}1$ or ${\boldsymbol{L}}2$, the outer ring can always grow to a proper K_o to impact on the inner ring and, according to the state selection of Figure 8, the two tori will collide before the outer ℓcorotating ring is accreting. In fact, there is ${r}_{\max }^{o}\lt {r}_{\max }^{i}\lt {r}_{\mathrm{mso}}\lt {r}_{\min }^{i}\lt {r}_{\min }^{o}\,,$ and then $\sup {y}_{3}^{o}={r}_{\max }^{o}\lt {r}_{\min }^{i}\lt {y}_{1}^{i}$. It is instantaneous to infer if both configurations are ${()}_{1}$. If, on the other hand, we have ${C}_{o}={C}_{2}^{o}$, then there is

$$\begin{eqnarray}&&{r}_{\gamma }\lt {r}_{\max }^{o}\lt {r}_{\mathrm{mbo}}\leqslant \sup {y}_{3}^{o}={\bar{{\mathfrak{r}}}}_{1}^{o}\lt {r}_{\max }^{i}\leqslant {y}_{3}^{i}\lt {r}_{\min }^{i}\,,\\ &&\quad {\rm{where}}\quad {\bar{{\mathfrak{r}}}}_{1}^{o}:\,{V}_{\mathrm{eff}}({{\ell }}_{o},{\bar{{\mathfrak{r}}}}_{1}^{o})=1,\quad {\bar{{\mathfrak{r}}}}_{\mathrm{mbo}}\lt {r}_{\min }^{o}\lt {\bar{{\mathfrak{r}}}}_{\gamma },\end{eqnarray} \tag{ 38 }$$

$$\begin{eqnarray}&&{\rm{if}}\quad {C}_{i}={C}_{2}^{i}\quad {\rm{then}}\,{\rm{there}}\,{\rm{is}}\,{\rm{also}}\,{\rm{a}}\\ &&\quad {K}_{i}:\,{\bar{{\mathfrak{r}}}}_{\mathrm{mbo}}\lt {r}_{\min }^{i}\lt {y}_{1}^{i}\leqslant {y}_{3}^{o}\lt {r}_{\min }^{o}\lt {\bar{{\mathfrak{r}}}}_{\gamma },\end{eqnarray} \tag{ 39 }$$

$$\begin{eqnarray}&&{\rm{if}}\quad {C}_{i}={C}_{1}^{i}\quad {\rm{then}}\,{\rm{there}}\,{\rm{is}}\,{\rm{also}}\,{\rm{}}\\ &&\quad {r}_{\gamma }\lt {r}_{\max }^{o}\lt {r}_{\mathrm{mbo}}\lt {r}_{\max }^{i}\lt {r}_{\mathrm{mso}}\lt {r}_{\min }^{i}\lt {\bar{{\mathfrak{r}}}}_{\mathrm{mbo}}\\ &&\quad \lt {r}_{\min }^{o}\lt {\bar{{\mathfrak{r}}}}_{\gamma }.\end{eqnarray} \tag{ 40 }$$

In Equation (38), we considered the fact that ${{\ell }}_{\mathrm{mbo}}$ is the inferior (in magnitude) of the ${\boldsymbol{L}}2$ range, and ${V}_{\mathrm{eff}}({{\ell }}_{\mathrm{mbo}},{r}_{\mathrm{mbo}})=1$, but $1={V}_{\mathrm{eff}}({{\ell }}_{\mathrm{mbo}},{r}_{\mathrm{mbo}})\lt {V}_{\mathrm{eff}}({{\ell }}_{o},{r}_{\mathrm{mbo}})$ for any ${{\ell }}_{o}\in ]{{\ell }}_{\mathrm{mbo}},{{\ell }}_{\gamma }[$. Therefore, there exists ${\bar{{\mathfrak{r}}}}_{1}^{o}\lt {r}_{\min }^{o}:\,{V}_{\mathrm{eff}}({{\ell }}_{o},{\bar{{\mathfrak{r}}}}_{1}^{o})=1\,\lt {V}_{\mathrm{eff}}({{\ell }}_{o}\,,{r}_{\mathrm{mbo}})$, which is, of course, possible if and only if ${\bar{{\mathfrak{r}}}}_{1}^{o}\in ]{r}_{\mathrm{mbo}},{r}_{\min }^{o}[$ (since ${\partial }_{r}{V}_{\mathrm{eff}}(r)\lt 0$ at $r\lt {r}_{\min }$). Moreover, we used the property that ${r}_{\mathrm{mbo}}^{\pm }\notin {C}_{j}^{\pm }$ for any ℓ_j.

Collision after growth or accretion of the innerring.In the argument above, we considered only the role of the outer configuration of a couple in the emergence of collision. It is, however, clear that in finding out the condition for ${y}_{1}^{i}={y}_{3}^{o}$, we should consider the couple of parameters $({K}_{i},{K}_{o})$. Particularly, we need to investigate the elongation of the inner torus in the equatorial plane, up to the extreme limit of the configuration ${{\rm{O}}}_{\times }$ or ${C}_{\times }^{1}$, eventually colliding with the outer ring. First we report here some immediate considerations that also hold for ℓcounterrotating couples. To fix the ideas, we consider an ℓcounterrotating couple $({C}_{i}^{-},{C}_{o}^{+})$ made by an inner corotating accreting torus and an outer counterrotating accreting torus as in Figure 11, with maximum elongation ${\lambda }_{\times }$ in the equatorial plane and the radii ${\bar{{\mathfrak{r}}}}_{\max }^{{+}_{i}}\lt {\bar{{\mathfrak{r}}}}_{\max }^{{+}_{o}}$, and solutions of ${V}_{\mathrm{eff}}({{\ell }}_{+},{r}_{\max }^{+})\equiv {K}_{\max }^{+}={\bar{K}}_{\max }^{+}$ for ${{\ell }}_{+}\in {\boldsymbol{L}}{1}_{+}$, where ${\bar{{\mathfrak{r}}}}_{\max }^{{+}_{i}}$ is the accretion point and ${\bar{{\mathfrak{r}}}}_{\max }^{{+}_{o}}$ is the outer edge of the counterrotating torus. The existence of the double system ${C}_{\times }^{-}\lt {C}_{\times }^{+}$ is ensured by the condition ${\bar{{\mathfrak{r}}}}_{\max }^{{+}_{o}}\leqslant {\bar{{\mathfrak{r}}}}_{\max }^{-}$ where the equality holds as a condition for collision as shown, for example, in Figure 7. Then, considering ${\bar{{\mathfrak{r}}}}_{\max }^{-}:{V}_{\mathrm{eff}}({{\ell }}_{-},{r}_{\max }^{-})\equiv {K}_{\max }^{-}={\bar{K}}_{\max }^{-}$, we need to choose $({\bar{K}}_{\max },{\bar{K}}_{\max }{]}^{-})$ on the curves ${V}_{\mathrm{eff}}({{\ell }}_{\pm },r)$, which is always possible to find as we can have $-{{\ell }}_{+}\lt -{\bar{{\ell }}}_{+}:{\bar{{\mathfrak{r}}}}_{\max }^{{+}_{o}}={r}_{\mathrm{mso}}^{-}$ and then $-{{\ell }}_{+}\in ]-{{\ell }}_{\mathrm{mso}}^{+},-{\bar{{\ell }}}_{+}[$ (Pugliese & Stuchlík 2016a).

Focusing on the couples ${C}_{\times }^{1}\lt {C}_{o}$ and using Equation (37), we find

$$\begin{eqnarray}&&{K}_{i}\in ]{K}_{\min }^{i},{V}_{\mathrm{eff}}({{\ell }}_{i},{r}_{\min }^{o})]\quad {\rm{and}}\\ &&\quad {K}_{o}={V}_{\mathrm{eff}}({{\ell }}_{o},{y}_{1}^{i})\gt {K}_{\min }^{o}\quad {\rm{for}}\,{\rm{}}\quad {{\ell }}_{i}{{\ell }}_{o}\gt 0,\end{eqnarray} \tag{ 41 }$$

$$\begin{eqnarray}&&{\rm{or}}\quad {K}_{\min }^{i}\lt {K}_{i}={V}_{\mathrm{eff}}({{\ell }}_{i},{y}_{1}^{i})\leqslant {V}_{\mathrm{eff}}({{\ell }}_{i},{r}_{\min }^{o})\\ &&\quad \lt {V}_{\mathrm{eff}}({{\ell }}_{o},{r}_{\min }^{o})={K}_{\min }^{o}\lt {V}_{\mathrm{eff}}({{\ell }}_{o},{y}_{1}^{i})={K}_{o}\lt {K}_{\max }^{o}\,,\end{eqnarray} \tag{ 42 }$$

and assuming an unstable inner configuration, Equation (41) implies

$$\begin{eqnarray}&&{K}_{\min }^{i}\lt {K}_{i}={K}_{\max }^{i}\leqslant {V}_{\mathrm{eff}}({{\ell }}_{i},{r}_{\min }^{o})\\ &&\qquad \lt {K}_{\min }^{o}\lt {V}_{\mathrm{eff}}({{\ell }}_{o},{y}_{1}^{i})={K}_{o}\lt {K}_{\max }^{o},\end{eqnarray} \tag{ 43 }$$

confirming that $({)}_{\times }^{i}\ne {{\rm{O}}}_{\times }^{2}$ (Pugliese & Stuchlík 2015). However, the necessary condition, Equation (43), and particularly the relation ${K}_{\max }^{i}\lt {K}_{\min }^{o}$ are satisfied only in some special cases. The following two cases may occur:

$$\begin{eqnarray}&&{\rm{I}}:\,\,{K}_{\min }^{i}\lt {K}_{\max }^{i}\lt {K}_{\min }^{o}\lt {K}_{\max }^{o}\quad {\rm{or}}\\ &&\,{\rm{\Delta }}{K}_{\mathrm{crit}}^{i}\lt {\rm{\Delta }}{K}_{\mathrm{crit}}^{o}\quad {\rm{and}}\quad {\rm{\Delta }}{K}_{\mathrm{crit}}^{i}\cap {\rm{\Delta }}{K}_{\mathrm{crit}}^{o}=0,\end{eqnarray} \tag{ 44 }$$

$$\begin{eqnarray}&&\begin{array}{l}\mathrm{II}:{\,K}_{\min }^{i}\lt {K}_{\min }^{o}\lt {K}_{\max }^{i}\lt {K}_{\max }^{o}\quad {\rm{or}}\\ \,{\rm{\Delta }}{K}_{\mathrm{crit}}^{i}\cap {\rm{\Delta }}{K}_{\mathrm{crit}}^{o}={K}_{\min }^{o}.\,\,\end{array}\end{eqnarray} \tag{ 45 }$$

The first follows Equation (41) and therefore satisfies the necessary condition for collision; the second forbids any collision after instability of the inner ring (which is preceded by merging). The two cases are regulated by the ratio of the specific angular momenta of the tori of the couple. Thus, there has to be a specific angular momentum ratio ${\ell }{/}_{c}\equiv {{\ell }}_{o}/{{\ell }}_{i}:{K}_{\min }^{o}={K}_{\max }^{i}\,,$ which is the discriminant case between Equations (44) and (45)—see Pugliese & Stuchlík (2015). The discriminant case has to satisfy the relation ${V}_{\mathrm{eff}}({{\ell }}_{i},{r}_{\max }^{i})={V}_{\mathrm{eff}}({{\ell }}_{o},{r}_{\min }^{o}\,)={V}_{\mathrm{eff}}({{\ell }}_{i},{y}_{1}^{i})$. However, in Pugliese & Stuchlík (2015), it was shown that in the case ${K}_{\max }^{i}={K}_{\min }^{o}\,,$ the outer ring ${C}_{o}$ cannot be unstable and the inner ring ${C}_{i}$ cannot be accreting. This is because if ${\lambda }^{i}={\lambda }_{\times }^{i}$, then there is ${y}_{1}^{i}={y}_{\min }^{o}$. In conclusion, the ringed disk cannot be unstable according to a P-W instability; neither can it form a ${{\boldsymbol{C}}}_{\odot }^{2}$ with an inner accreting torus. Then the two rings collide before the inner disk will reach the accretion phase. At fixed ℓ_o, the specific angular momentum magnitude for this case is ${{\ell }}_{i}={\ell }{/}_{i}\equiv {{\ell }}_{o}/{\ell }{/}_{c}\in {\boldsymbol{L}}1$ or ${\ell }{/}_{c}{{\ell }}_{\mathrm{mso}}\lt {{\ell }}_{o}\lt {\ell }{/}_{c}{{\ell }}_{\mathrm{mbo}}$. As ${\ell }{/}_{c}\gt 1$ by definition, this case will hold for a part of ${C}_{2}^{o}$ and at least for a part of ${C}_{1}^{o}$ and possibly a ${C}_{3}^{o}$ tori, according for the condition ${{\ell }}_{o}\gt {{\ell }}_{i}({{\ell }}_{\gamma }/{{\ell }}_{\mathrm{mbo}})$. This last condition distinguishes the ℓcorotating couples of corotating or counterrotating rings where ${{\ell }}_{\gamma }^{+}/{{\ell }}_{\mathrm{mbo}}^{+}\gt {{\ell }}_{\gamma }^{-}/{{\ell }}_{\mathrm{mbo}}^{-}$ and ${\partial }_{a}({{\ell }}_{\gamma }^{\pm }/{{\ell }}_{\mathrm{mbo}}^{\pm })\gtrless 0$. Then the necessary condition for colliding ${{\boldsymbol{C}}}_{\odot }^{\times }$ for ${C}_{\times }^{1}$ may be rephrased by saying that ${{\ell }}_{i}\lt {\ell }{/}_{i}$ in magnitude. Consistently, the condition in Equation (45) also prohibits also the inner torus to be in accretion. Focusing on the discriminant case, with reference to Figure (12), we consider the angular momenta associated with this case. We find the solutions of the problem ${K}_{\mathrm{crit}}^{\pm }\,=$ constant for ℓcorotating couples of counterrotating or corotating tori, respectively. The solution provides the angular momenta related to the constant surfaces of the curve K_crit, as functions of the constant value $c\geqslant {K}_{\mathrm{mso}}^{\pm }$ in the range ${K}_{\mathrm{crit}}\in {\boldsymbol{K}}0$, say $\pm {{\ell }}_{c}^{\gt }{}^{\mp }\geqslant \pm {{\ell }}_{c}^{\gt }{}^{\mp }$. The two panels are to be read as follows: the (horizontal) lines ${K}_{\mathrm{crit}}\,=$ constant in the first panel show ${K}_{\mathrm{crit}}\,={K}_{\max }^{i}={K}_{\min }^{o}$ and the two associated radii ${r}_{\max }^{i}\lt {r}_{\min }^{o}$, for the two ℓcorotating configurations with unknown $\pm {{\ell }}_{\mp }^{i}\lt \pm {{\ell }}_{\mp }^{o}$. Symbols on the curve indicate the K_crit of the couple associated with equal ${\ell }$ and therefore to one ring. On the other hand, we could, as done in Pugliese & Stuchlík (2015, 2016a), use the curve ${\ell }(r)$ to find out $({{\ell }}_{i},{{\ell }}_{o})$, through the r_crit obtained from this first panel; however, here we can get this information alternatively by using the second panel of Figure 12. Then, by taking this K_crit on the second panel (vertical line), we select the two angular momenta ${{\ell }}_{i}={{\ell }}_{c}^{\lt }$ and ${{\ell }}_{o}={{\ell }}_{c}^{\gt }$, respectively, associated with the two ℓcorotating rings with ${K}_{\max }^{i}={K}_{\min }^{o}\,.$ Thus, on this second panel the (horizontal dashed) lines ${{\ell }}_{i}={{\ell }}_{c}^{\lt }$ = constant and ${{\ell }}_{o}={{\ell }}_{c}^{\gt }\,=$ constant, crossing the curve ${{\ell }}_{c}^{\gt }$ and ${{\ell }}_{c}^{\lt }$, respectively, set the couples ${K}_{\min }^{i}$ and ${K}_{\max }^{o}$ (details with configurations with special K can be found in Pugliese & Stuchlík 2015).

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Investigating the topology of the couple, we note that ${{\ell }}_{c}^{\lt }$ is also well-defined for $K\in {\boldsymbol{K}}1$; therefore ${\ell }\in {\boldsymbol{L}}2$, and it is associated with the instability points (in the region $r\lt {r}_{\mathrm{mso}}$). The curve associated with the minimum points, ${{\ell }}_{c}^{\gt }$ (region $r\gt {r}_{\mathrm{mso}}$), instead correctly extends only in ${\boldsymbol{K}}0$ up to ${\boldsymbol{L}}3$, confirming that the discriminant case occurs only in ${\boldsymbol{K}}0$. Fixing ${K}_{\mathrm{crit}}={\bar{K}}_{\mathrm{crit}}$, a torus ${C}_{\odot }^{\times }$ can then be formed if ${{\ell }}_{o}={{\ell }}_{c}^{\gt }\gt {\bar{{\ell }}}_{c}^{\gt }$ or ${{\ell }}_{i}={{\ell }}_{c}^{\lt }\lt {\bar{{\ell }}}_{c}^{\lt }$, where the values ${\bar{{\ell }}}_{c}^{\gtrless }$ are associated with the line ${K}_{\mathrm{crit}}={\bar{K}}_{\mathrm{crit}}=$ constant. More precisely, we obtain, at fixed ${{\ell }}_{o}={\bar{{\ell }}}_{c}^{\gt }$ (and a proper choice of K_o), collision after accretion of the inner ring only if its specific angular momentum is small enough in magnitude i.e., ${{\ell }}_{i}={{\ell }}_{c}^{\lt }\lt {\bar{{\ell }}}_{c}^{\lt }$, while for larger values of the magnitude of the specific angular momentum, ${{\ell }}_{i}={{\ell }}_{c}^{\lt }\geqslant {\bar{{\ell }}}_{c}^{\lt }$, for any ${K}_{o}\in ]{\bar{K}}_{\mathrm{crit}},{K}_{\max }^{o}[\subset {\boldsymbol{K}}0$ (where ${K}_{\max }^{o}:\,{\bar{{\ell }}}_{c}^{\gt }={{\ell }}_{c}^{\lt }$ if ${\bar{{\ell }}}_{c}^{\gt }\in {\boldsymbol{L}}1$ or we can take ${K}_{\max }^{o}\equiv 1$ if ${\bar{{\ell }}}_{c}^{\gt }\in {\boldsymbol{Li}}$ with ${\boldsymbol{Li}}\in \{{\boldsymbol{L}}2,{\boldsymbol{L}}3\}$ as we are interested only in the closed ${C}_{o}$ equilibrium configurations), the possible phase of accretion of the inner torus must be preceded by the collision and possibly the merging of the two rings. Collision by accretion, for small enough K_o, can take place only for ${{\ell }}_{i}\lt {\bar{{\ell }}}_{c}^{\lt }{}^{-}$. We can therefore also provide an upper boundary for the angular momentum of the inner surface as ${\bar{{\ell }}}_{c}^{\gt }\in {\boldsymbol{L}}1$. In fact as is clear from Figure 1, when the angular momentum of the inner torus reaches this limit, then the two tori overlap completely $({r}_{\min }^{i}={r}_{\min }^{o}\,),$ if ${\bar{{\ell }}}_{c}^{\gt }\in {\boldsymbol{L}}1$. Whereas if ${\bar{{\ell }}}_{c}^{\gt }\in {\boldsymbol{Li}}\gt {\boldsymbol{L}}1$, then ℓ_i is bounded by ${{\ell }}_{\mathrm{mbo}}$ from above. This trend is qualitatively independent of the spin of the attractor and the direction of rotation of the tori with respect to the attractor. Finally, condition (43) may also hold for a ${C}_{3}^{o}$ ring, because the maximum of the outer configuration is not actually involved. Therefore, the problem will in turn be how small should ℓ₃ be for the elongation of the inner ring in accretion to match the outer ring. Further discussions regarding the possible loops in the double systems are in Appendix A.1.

The systems investigated here and in Pugliese & Stuchlík (2015, 2016a) offer a methodological challenge of describing a set of virtually separated subsystems as an entire configuration. The double tori of the ringed accretion disk may have different topologies and geometries characterized by different rotation laws, giving rise to four different spin–spin alignments with respect to the spin of the central attractor, as sketched in Figure 4. The evolution of the entire macro-configuration would then result from the evolution of each subconfiguration, which reaches an interacting phase when the two configurations eventually reach contact. Tori in a double system may collide and merge, or, eventually, turn to generate some feeding–drying processes: the accreting matter from the outer torus of the couple can impact on the inner torus, or the outer torus may be inactive with an active inner torus accreting onto the BH, or both tori may be active. We demonstrate that some configurations will collide for some initial conditions and dimensionless spins of the attractors. This process likely ends in the formation of a single orbiting toroidal accretion disk. Our studies may also provide information on the SMBH accretion disk formation due to BH interaction with the environment in different stages of its life. The phenomenology associated with these systems may therefore be very wide, and we believe that this study could open up a new field of investigation in astrophysics, leading, as also proposed by Sochora et al. (2011), Karas & Sochora (2010), and Schee & Stuchlik (2009), to the reconstruction of the interpretive framework of some phenomena in AGN environments, which have so far been thought of in terms of a single accretion disk, in terms of multiple accretion disks. The inner edge of the outer ring and the role played by the outer edge of the inner disk should be crucial.

As discussed in Pugliese & Stuchlík (2016a), a strong overflow of matter from a torus of the double configuration could be also related to jet formation. Emission may be released in high energy collisions. The enormous energy emitted by the accretion disks in quasars or AGNs, in the form of electromagnetic radiation and jets, is generally attributed to the strong gravity of the central BH when the gravitational binding energy of accreting matter is transformed into radiation.

Finally, we stress that the study of the equilibrium tori could be the starting point for a future analysis of the oscillation modes in the structure of relativistic ringed disks, which can be related to various astrophysical phenomena. The radially oscillating tori of the ringed disk could be related to the high-frequency quasi-periodic oscillations observed in non-thermal X-ray emission from compact objects (QPOs), a still obscure feature of X-ray astronomy related to the inner parts of the disk.

We investigated evolutionary schemes of ringed accretion disks constituted by two toroidal axisymmetric tori orbiting on the equatorial plane of a central supermassive Kerr BH. We discussed the emergence of the instability phases for each ring of the macro-configuration in full general relativistic treatment by considering the effects of the geometry of the Kerr spacetimes on the systems. As results of this analysis, we identified particular classes of central Kerr attractors depending on their dimensionless spins and the constraints imposed on the evolutionary schemes of the double toroidal system. The schemes outline the topological transition of the tori from an equilibrium topology to instability depending on the rotation of the tori relative to each other and to the central Kerr BH. States representing the pair of tori for the four macro-configurations listed in Figure 7 are summarized in Table 2. We used these blocks to construct the evolutionary schemes in Figure 8 for a couple of tori in the Schwarzschild spacetime and the ℓcorotating pairs in a Kerr spacetime, while Figure 9 shows the case of ℓcounterrotating pairs orbiting a Kerr attractor. These couples may be formed only in certain stages of the inner torus evolution for some Kerr attractors. Our analysis in turn sets significant limits on the observational evidence of these systems, providing constraints on the tori–attractor systems, imposing limits on the central attractor spins, and relating the attractor to the torus couple and their evolution from formation to final stage toward accretion or collision. The presented analysis of evolutionary schemes is related to the case of "frozen" Kerr geometry. We here do not follow the evolution of the (M, a) parameters of Kerr spacetime. In some cases, inclusion of the evolution of the a/M parameter could introduce some instabilities when some critical values of a/M will lead to accretion of corotating or counterrotating matter. The mutation of the geometry determines in general a change of the dynamical properties of the tori, eventually resulting, as argued in Pugliese & Stuchlík (2015), in an iterative process, which could even give rise to runaway instability (Abramowicz et al. 1983, 1998; Font & Daigne 2002a; Rezzolla et al. 2003; Hamersky & Karas 2013; Korobkin et al. 2013; Pugliese & Quevedo 2015). This is a theme for future work. Finally, we considered the situation in which the pair of tori may collide, either remaining quiescent or after the emergence of instability from one of the subconfigurations, discussing the mechanisms that, during tori evolution, could lead to collision. The possible scenarios may eventually end in the merging of the tori with the destruction of the macro-configuration. Note that in the case of ℓcorotating tori, the outer torus collides with the inner torus; eventually the tori merge before it can reach the accretion phase, with the consequent destruction of the system (therefore it could not give rise to a configuration ${{\boldsymbol{C}}}_{\odot }^{x}{}^{2}$ with accretion point ${r}_{\times }={y}_{1}^{i}={y}_{3}^{o}$). Similarly, according to the double geodesic structure of the Kerr spacetime, there is no outer corotating torus in accretion in a couple with an inner counterrotating torus, as discussed in Section 3.2.1 for the couples ${()}^{+}\lt {()}^{-}$.

Our interest in this investigation was justified by a series of studies and observational evidences supporting the existence of supermassive BHs characterized by multi-accretion episodes during their lifetime. These facts justify questioning the relevance of ringed accretion disks theoretically and challenging them phenomenologically. The presence of such structures substantially modifies the thus-far assumed scenario of a single disk, taken as the basis of High Energy Astrophysics, connected with accretion. New observational effects may then be associated with these complex structures, thus showing their existence unequivocally. We have different pieces of evidence suggesting that such a situation might be the case. From a theoretical perspective, there are a number of possible physical mechanisms for ringed disk formation. Our work is related to their dynamics around SMBHs, where curvature effects become relevant, and the general relativistic treatment adopted here is the most appropriate. There are indeed suggestions for these objects to be hosted in the geometries of SMBHs; therefore we assume these systems to be the most probable environments. It is generally accepted then that the nuclei of most galaxies contain SMBHs; this picture is supported by several studies and agrees with several observational facts. In these environments, the history of SMBHs may show traces of its host galaxy dynamics. Repeated galaxy mergers may constitute one mechanism for diversified feeding of an SMBH. Probably, the more immediate situation to think of where a ring of matter may be formed is in binary BH systems, and this applies in many astrophysical contexts: X-ray binaries or SMBH binary systems, which are characterized by diverse accretion episodes, feeding BHs with matter and angular momentum. Concerning then the possibility of counterrotating disks, which we fully address in the present paper, we refer to well-known studies providing strong evidence and a further fascinating hypothesis of misaligned disks. On this last possibility, there is quite a large body of literature; we refer, for example, to Aly et al. (2015), Dogan et al. (2015), and Lodato & Pringle (2006). Another mechanism, perfectly fitting in particular with our model of ℓcorotating tori, foresees a splitting of one accretion disk, resulting in the formation of a torus couple. In other words, a ringed accretion disk may result from a fragmentation of a previously single accretion disk, due to local self-gravitational instability. All these facts lead us to support the suggestion that the existence of such structured objects are likely to be considerably significant in AGNs. Therefore, from the observational point of view, we expect our results have implications for a number of different observational features of AGNs, and we have marked some before. On the other hand, it is opening possibilities for different observational evidence for new intriguing phenomena induced by the tori interactions or oscillations. The presence of an inner tori may also enter as a new unexpected ingredient in the accretion–jet puzzle. From a methodological viewpoint, the study of such systems clearly opens an incredible amount of possibilities to be investigated. The analysis introduced here shows the huge number of cases that occur even within a simple three-parameter model (the specific angular momentum ℓ, the K parameter, and the attractor spin–mass ratio). Thus, we now have two possibilities for approaching the analysis: by numerically solving in diversified scenarios the equations for a very specific case and fixing the disk and attractor parameters. In this way we can also include more ingredients in each disk model, but we lose the general overview of the situation, needing moreover to set some initial configurations. The investigation developed in this work fixes these two issues: we substantially reduce the parameter space of our model, providing a range of variations of the variables and parameters that may also fit to some extent other disk models, and also providing attractor classes on the bases of the features of the tori, ultimately indicating the attractor that we should chase to find evidence. We were also able to provide indication of the initial disk couple evolution. Any numerical analysis of more complex situations, sharing the same symmetry of at least one disk, should be compelled by the results presented here.

In particular, this study (Section 3.3) provides strong constraints for the model parameters of the evaluation of the center-of-mass energy in collisions between rings, which was first evaluated, within the test particle approximation, in D. Pugliese & Z. Stuchlik (2017, in preparation). It was proven that the energy efficiency of the collisions increases with increasing dimensionless BH spin, giving very high values for near-extreme black holes—such systems can be significant for the high energy astrophysics related especially to accretion onto supermassive black holes and the extremely energetic phenomena in quasars and AGNs.

In conclusion, we believe our results may be of significance for high energy astrophysical phenomena, such as the shape of X-ray emission spectra, the X-ray obscuration and absorption by one of the rings, and the extremely energetic radiative phenomena in quasars and AGNs that could be observable by the planned X-ray observatory ATHENA.¹⁰

D.P. acknowledges support from the Junior GACR grant of the Czech Science Foundation No:16-03564Y. Z.S. acknowledges the Albert Einstein Center for Gravitation and Astrophysics supported by Czech Science Foundation Grant No. 14-37086G. The authors have benefited during the preparation of this work from discussion with a number of colleagues. In particular, we thank Prof. J. Miller, Prof. M. A. Abramowicz, and Prof. V. Karas. We would also like to thank the anonymous reviewer for the useful suggestions and constructive comments, which helped us to improve the manuscript.

In this section, we clarify some aspects of the graphs formalisms used in this work. State lines of the ℓcorotating couples, used in the monochromatic graph of Figure 8, are listed in Figure 14, while the state line for the ℓcounterrotating couples, used in the bichromatic graphs of Figure 9, are listed in Figure 13. Samples of loop graphs discussed in this work are in Table 15. The evolutive lines generally split the graph into two parts, centered around the center: the antecedent section, from which the heads of the arrows converging at the center start, and the subsequent section, which is the one onto which the evolutive lines, starting from or crossing the graph center, converge, with the head in the antecedent section—Table 15. A graph may also have only one section. As pointed out in Section 3, the evolutive lines may connect state lines with different critical sequentialities but not different configuration sequentialities, which is preserved during the evolution. In this discussion, vertices that are connected or crossed by an evolutive line, pertain to equal chromaticity. Thus, if the graph center is monochromatic (bichromatic), then the entire graph is monochromatic (bichromatic). An evolutive loop is defined as the union of evolutive lines and the vertices they cross, closing on an initial topology as in Table 15, The triple vertex transition is an example of a loop .

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A.1. Consideration of the Loop Emergence

We provide some comments on Table 2. The analysis refers to Figure 2, where the complementary geodesic structure ${\bar{{\boldsymbol{R}}}}_{{\rm{N}}}$ is represented.

B.1. Couples: ${()}^{+}\lt {()}^{-}$

B.2. Couples: ${()}^{-}\lt {()}^{+}$

B.3. Comments on the Constrained Criticality

Here we provide some notes on the results of Table 2 on the criticality order, discussing also the order of decorated state lines for the counterrotating couples of Figure 14. These constraints on the location of the instability points are consequences of the geodesic structure of spacetime, as represented in Figure 1. However, we considered also restrictions provided by Pugliese & Stuchlík (2016a), based on the geodesic structure ${\bar{{\boldsymbol{R}}}}_{{\rm{N}}}$ and the relation between the angular momenta ${{\ell }}_{{\rm{N}}}$—Figure 1. For $a\gt {a}_{{\iota }_{a}}$, the geodesic structure implies ${C}_{\times }^{-}\,\prec \,{C}_{\times }^{+}$, where ${a}_{{\iota }_{a}}=0.372583M:\,{r}_{\mathrm{mbo}}^{+}={r}_{\mathrm{mso}}^{-}$, where the critical sequentiality is not fixed. However, it is possible to prove that the relation ${C}_{\times }^{-}\,\prec \,{C}_{\times }^{+}$ extends also for couples orbiting around slower attractors. In fact, for $a\lt {a}_{{\iota }_{a}}$, there is ${r}_{\mathrm{mbo}}^{-}\,\lt {r}_{\mathrm{mbo}}^{+}\lt {r}_{\mathrm{mso}}^{-}\lt {r}_{\mathrm{mso}}^{+}$. However, ${r}_{\mathrm{mbo}}^{+}\in !{C}_{\times }^{-}$, and ${r}_{\mathrm{mso}}^{+}\notin {C}_{\times }^{-}$—this means in particular that ${r}_{\mathrm{mbo}}^{-}\,\lt \,{y}_{3}^{-}\,\lt \,{r}_{\mathrm{mbo}}^{+}\lt {y}_{3}^{+}\lt {r}_{\mathrm{mso}}^{-}\lt {r}_{\mathrm{mso}}^{+}$. Therefore, ${C}_{\times }^{-}\,\prec \,{C}_{\times }^{+}$, which closes the proof. Note that we used the assessment of ${r}_{\mathrm{mbo}}^{+}\in !{C}_{\times }^{-}$ obtained from the considerations in Pugliese & Stuchlík (2016a). Moreover, ${r}_{\mathrm{mso}}^{-}\,\in /\,{C}_{\times }^{+}$ if ${{\ell }}_{1}\in ]{{\ell }}_{\mathrm{mso}}^{+},{{\ell }}_{1}({r}_{\mathrm{mso}}^{-}),[\,{r}_{\mathrm{mso}}^{-}\in !{C}_{\times }^{+},\,{{\ell }}_{1}\in ]{{\ell }}_{1}({r}_{\mathrm{mso}}^{-}),{{\ell }}_{\mathrm{mbo}}^{+}[$. A similar analysis could be implemented for other topologies. For $a\gt {a}_{\iota }$, where ${a}_{\iota }=0.313708M:\,{r}_{\gamma }^{+}={r}_{\mathrm{mbo}}^{-}$, we have ${{\rm{O}}}_{\times }^{-}\,\prec \,{{\rm{O}}}_{\times }^{+}$, while ${{\rm{O}}}_{\times }^{-}\,\prec \,{C}_{\times }^{+}$ in any geometry. Finally, for $a\gt {a}_{{\gamma }_{+}}^{-}\,=0.638285M:\,{r}_{\mathrm{mso}}^{-}={r}_{\gamma }^{+}$, ${C}_{\times }^{-}\,\prec \,{{\rm{O}}}_{\times }^{+}$.

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