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On the dynamics of the torus around the kicked black hole

There is a special interest to understand the dynamical properties of the accretion disk created around the newly formed black hole due to the supermassive black hole binaries which merge inside the gaseous disk. The newly formed black hole would have a kick velocity at up to thousands of km/s that drives a perturbation on a newly accreted torus around the black hole. Some of the observed supermassive black holes at the center of the Active Galactic Nucleus (AGN) move with a certain velocity relative to its broader accretion disk. In this paper, the effects of the kicked black holes onto the infinitesimally thin accreted torus are studied by using the general relativistic hydrodynamical code, focusing on changing the dynamics of the accretion disk during the accretion disk-black hole interaction. We have found the non-axisymmetric global mode m=1 inhomogeneity, which causes a spiral-wave structure, is excited on the torus due to kicked black hole. The higher the perturbation velocity produced by the kicked black hole, the longer time the torus takes to reach the saturation point. The created spiral density waves which rapidly evolve into the spiral shocks are also observed from the numerical simulations. The spiral shock is responsible for accreting matter toward the black hole. Firstly, the spiral-wave structure is developed and the accretion through the spiral arms is stopped around the black hole. At the later time of simulation, the formed spiral shocks partly cause the angular momentum loss across the torus.


Introduction
In recent years, number of numerical simulations (Baker et. al. 2008;Kornreich & Lovelace 2008;Gerosa et. al. 2018) and astronomical observations (Webb et.al. 2018;Bustillo et. al. 2018;O'Shaughnessy et. al. 2017;Atri et. al. 2019) have revealed the kicked black hole at the center of the black hole X-ray binaries and the AGN. This has occurred due to the asymmetric momentum loss of the black hole, either in a case of black hole mergers or in the release of the gravitation radiation during this process. It was found from the numerical simulation that the gravitational wave recoil velocities range from the smallest possible value up to thousands of km/s (Lousto & Zlochower 2011;Ponce et. al. 2012). The supermassive black hole kick velocities were also measured from the smallest possible value up to 500km/s (Gerosa & Moore 2016). It is also expected to be a kicked velocity from the binary coalescence in the galaxy mergers. Kick velocities up to a few hundred km/s of the stellar mass black holes formed in the core-collapse supernovae were first described by Bekenstein (1973) which might account for their relatively small branching ratio to long GRBsvan Putten (2004).
A smoothed particle hydrodynamics numerical simulation of the impulsive kick transmitted to a black hole on the dynamical evolution of the accretion disk has been done by Ponce et. al. (2012). They have found the higher potential luminosity considering the region 0.4pc from the galactic center for more oblique kick angle. The occurrence of the kicked black hole inside the gaseous disks perturbs the surrounding gas and it could create a signature of the electromagnetic radiation which gives an evidence of merging. The kick directed into the equatorial plane was studied by Corrales et. al. (2010) using the 2D FLASH code. The one-armed spiral shock wave appeared as a characteristic respond and it produced a total luminosity up to 10% of the Eddington luminosity. The generated shocks' energy can produce electromagnetic flares that could last up to hundreds of thousands years (Shields & Bonning 2008;Schnittman & Krolik 2008).
One of the new observations was made in a distant galaxy which is known as quasar 3C 186 (Chiaberge et.al. 2017). It is the most massive black hole which is kicked out of its central home. The merging two black holes produces gravitational waves which carry energy and angular momentum of the system. Finally, when they merge in a violent collision, and the released energy is enough, the newly formed black hole would be kicked away from the center of the galaxy or even out of the entire galaxy in the opposite side of the location with the highest gravitational waves. The range of the kick can vary depending on the properties of these two colliding black holes, their separation, and inclination angles.
The more observations showed that the broad-line emission system has a different red-shift when it is compared with the narrow-line emission system in several AGNs, Chiaberge1. such as SDSSJ092712.65 + 294344.0 with a blueshift of 2650km/s (Komossa et. al. 2008), SDSS J105041.35 + 345631.3 with a kicked velocity 3500km/s (Shields et. al. 2009b), CXOC J100043.1 + 020637 with a kicked velocity 1200km/s (Civano et. al. 2010), and E1821 + 643 with a kicked velocity 2100km/s (Rabinson et. al. 2010). When the black hole is kicked, broadline emitting region is dragged out along with the kicked black hole while the narrow-line region is left behind (Chiaberge et.al. 2017). It is believed that Radio-loud AGN with strong relativistic jets could contain the rapidly spinning black holes. Fastly spinning black holes could lead the accreted matter getting close to the black hole horizon and create a strong interaction between the matter and black hole.
High energetic astrophysical sources such as Gamma Ray Bursts (GRBs) (Kumar & Zhang 2015), X-ray binaries (Remillard & McClintock 2006), and AGNs (Reynolds 2014) are powered by whether the accretion mechanism onto the black holes or the spin energy of the black hole (van Putten 2002). A high accretion rate could be presented during the development of the instability. One of those instability is created by gravitational instability (Rice et. al. 2003;Lodato & Rice 2005), which is prone the development of the non-axisymmetric instability, called spiral density waves. The time dependent rest-mass density causes the material to concentrate at certain places in the effective potential around the black hole and generates spiral arms. The arms slowly rotates and allow the the disk angular momentum transform outwards ( Blaes 2014). Spiral structure mechanism leads developing a global non-axisymmetric modes in torus. The accreted matter belongs to spiral arms are characterized by the high internal temperatures which cause a high-energy emission from such sources as AGNs, X−ray binary, and GRBs (Reynolds 2014;Remillard & McClintock 2006). During the development of spiral structure, some vigorous dynamics of the torus would be observed. The vigorous phenomena happening on the torus dynamics around the kicked black hole would cause the conversion of the gravitational binding energy into thermal and kinetic energies during the process of the losing the angular momentum.
The oscillating properties of tori around the black hole and neutron stars have been investigated theoretically and numerically by perturbing the system Abramowicz & Fragile 2013). The numerical simulations of the radially perturbed accretion torus around the rotating and non-rotating black hole are performed by (Lee et.al. 2004). The perturbation of the black hole torus system can be observed in different types of perturbations such as angular velocity perturbation (Donmez 2017), spherical shell accretion (Donmez 2015), radial velocity perturbation , and Bondi-Hoyle accretion , which are all studied to extract the oscillation properties of the torus' PPI instabilities, and the quasi-periodic frequencies produced during these oscillations. The non-axisymmetric perturbations trigger not only the PPI instability but also spiral waves and cause the excitation of frequencies due to the non-linear coupling of the modes. Doing numerical simulation enables us to study the response of the accreted torus if the kicked is directed into the equatorial plane of the accretion disk.
The first numerical results of perturbation of the initially stable torus due to the kicked black hole at the center of the system are reported. To this extent, we solve the general relativistic hydrodynamical equations in a fixed Kerr space-time metric to investigate how a torus reacts to the kicked black hole. The outline of the paper is as follows. The computational framework of the problem with the physical variables of stable torus, properties of the black hole, and the radial velocities assumed created by kicked black holes are given in Section 2. In Section 3, dynamical properties of the kicked torus found from the numerical simulations are provided around the rotating and the non-rotating black holes. The spiral wave mode m = 1 generated by gravitational instability is reported and the dynamical structure of perturbed torus around the rotating black hole with the non-rotating case is compared. Finally, the summary of our results is given in Section 4. Unless otherwise specified, throughout this paper, the geometrized unit, c = G = 1 has been used.

Computational Framework of the 2D Model
To model the effects of the kicked black hole on the stable torus, we solve the General Relativistic Hydrodynamical (GRH) equations on the equatorial plane. GRH equations are described for the relativistic perfect fluid in the spacetime coordinate which can be written in covariant and conservative forms. The covariant form of GRH equation is T ab = ρhu a u b + P g ab stress-energy-momentum tensor for perfect fluid. The tensor depends on the specific enthalpy h = 1 + + P ρ , the rest mass density ρ, pressure P , the four-velocity u a , and the four-matrix of the space-tiem g µν . The perfect fluid equation of state P = (Γ − 1)ρ is used to constrain the system. After using 3 + 1 formalism, Eq.1 can be written conservative form. The formulation of GRH equation and its numerical solution were discussed with the full details in Donmez (2004) (see also (Donmez 2006(Donmez , 2012). GRH equations are solved around the non-rotating and rotating black holes in curved space-time which is described by Kerr line element, where α and β i are the lapse function and shift vectors, respectively, which are and where A = r 4 + r 2 a 2 + 2M ra 2 . a is dimensionless black hole spin parameter and M is the mass of the black hole.
We model the torus' dynamic defined on the equatorial plane in the limit of an infinitesimal thin disk. In fact, the tickness of the disk is finite and icreasess during the accretion of the matter towards the black hole. The effect of the disk tickness on the non-axisymmetric instability, spectra, and the black hole spin would be small (Zhou et. el. 2020). It is assumed that the black hole is kicked during one of the astrophysical events processes. As a result of this kick, the oscillating black hole transfers the angular momentum to the accreted torus, and the stable torus is perturbed. In the case of kicked black hole, the matter around the black hole receives an additional velocity. It is handled by focusing on perturbing the radial velocity of the torus at every time step at the center of galaxies or AGNs. The radial perturbations of the initially stable relativistic torus on the equatorial plane around the black holes due to the kicked black hole at the center of galaxies are studied by updating the radial velocity of the torus with the following expressions: where v r new and v r old are the velocities of torus' matter in the current and previous time steps respectively, and the amplitude of the oscillation is controlled by χ. Hence, low and high velocity kicks are controlled by changing this parameter.
The non-self-gravitating relativistic torus was first studied analytically by Abramowicz et. al. (1978). A sharp cusp was found for marginally stable accretion disk. The disk was located on the equatorial plane around the black hole. General relativistic torus with a perfect fluid equation of state was numerically discussed in  ;Nagar et. al. (2005). They computed the initial values of stable accreting tori around the nonrotating and the rotating black hole using Γ = 4/3 to mimic a degenerate relativistic electron gas. In order to have a system in a hydrodynamical equilibrium, the centrifugal and gravitational forces are balanced by the internal pressure in the torus. In this paper, we study the non-stable black hole torus system by perturbing the initially stable and non-self-gravitating relativistic tori which were given in  ;; . Initial relativistic tori used in our numerical simulations are almost stable during the evolution without a perturbation as seen in Figs.9 and 10 in . Initial physical parameters of the stable torus and kicked parameters used throughout this paper are summarized in Table 1.
The initial setups of tori around the non-rotating and the rotating black holes are thin, nonself-gravitating, and non-axisymmetric and defined on the fixed space-time metric using the Kerr coordinate. The disk-to-black hole mass ratio chosen as Mt M = 0.1 in our initial models makes the torus-black hole system stable with respect to the runaway instability. Therefore these models only respond to perturbations with a small amplitude which creates instabilities on the torus (Daigne & Font 2004;. Although the mass ratios for the real accreting systems are significantly lower such as X−ray binaries and AGNs (van Putten & Gupta 2009), the instabilities created after the applied perturbation could show the same behavior due to the following reasons. First, the self gravity of the torus is neglected. Second, the fixed background metric will not be affected due to the falling matter into a black hole. Finally, our torus is located very close to the black hole horizon. The gravitational time scales would be much larger than the other time scales in our numerical domain. The 2D perturbation of the relativistic torus on the equatorial plane is modeled in a polar coordinate. The inner and outer boundaries of the computational domain are extended from r in = 2.8M to r out = 100M around the non-rotating black hole. It goes from r in = 1.7M to r out = 100M for the rotating black hole. The uniform grid is used in the all models along the radial and the angular directions with N r = 1024 × N φ = 512 cells. The outflow boundary conditions are imposed at inner and outer boundaries of computational domain. The time evolution analysis of the stable torus on a fixed space-time metric without any perturbation shows that the structure of initial torus never changes ( does not develop a instability) during the simulation using the Cowling simulation (Cowling 1934). The neglecting the perturbation of the gravitational field is known as Cowling approximation. The Cowling approximation of the general relativistic system can be satisfied by setting the perturbation to be zero in the metric (Yoshida & Kojima 1997). The cowling approximation guarantees having a unperturbed gravitational field. The negligible mass accretion rates freeze the growth of instability oscillation mode in the Cowling approximation (Hawley 1991).
Main parameters of the initial stable torus around the black hole and its perturbation are given in Table 1, including the model name, black hole spin parameter a, torus-to-hole mass ratio M t /M , the maximum density of the torus ρ c (geo), constant specific angular momentum 0 , cusp location r cusp (M ). r c and r lso (M ) represent the location of the maximum value of density and the inner radii of the last stable orbit respectively, and t orb is the orbital period at r = r c . And χ(geo) is a free parameter to control the perturbation on the torus radial velocity. The adiabatic index Γ = 4/3, the polytropic constant K(geo) = 4.969 x 10 −2 is used in all simulations. We have used some negligible values for the initial density and pressure with a zero radial and angular velocities outside the torus. Mainly, we have constructed two different initial stable tori. First one is the torus around the rotating black hole and the second one is around the non-rotating black hole. The constant polytropic index and specific angular momentum are used in all our models.
The characterization and analysis of the instability are done by using the Fourier mode analysis explained in . Mainly, we calculate the power mode by computing the real and imaginary parts in Fourier transfer in order to analyze the azimuthal mode m.

Numerical Results
In this work, we have concentrated on a black hole-torus system which was perturbed by a kicked black hole, believed to occur at the center of AGN, and have conducted the dynamical calculations of the kicked torus. We neglect the self-gravity of the torus material and magnetic field along with using fixed space-time metric around the rotating and the non-rotating black holes. The immediate change in the kinetic energy of the torus' matter due to the kick would adjust the new equilibrium configuration. The change would also cause the transferring angular momentum of the torus which liberates the gas falling toward or away from the black hole. The redistribution of the angular momentum of the accreted torus due to the oscillating black hole allows the newly formed structures of the disk to evolve. During the evolution, the mass would be transferred, which can lead to forming an spiral arms and a quasi-periodically oscillation of the disk.

Rotating Black Hole
In order to study the effect of the kicked black hole onto the resulting torus dynamical evolution, we perform the number of numerical simulations around the rotating black hole for the  Morphology and dynamical evaluation of perturbed torus around the rotating black hole for model K09C is given in Fig.1. This figure represents the evaluation of the logarithmic rest-mass density on the equatorial plane at different snapshots. The last snapshot given in the figure shows the distribution of matter after the matter has almost completed 446 orbital periods around the black hole. The orbital period of the models for each model is given in Table 1. As it is seen from the snapshots, perturbation due to the kicked black hole initially triggers the spiral arms around the equilibrium. The instability is extended all over the torus surface. The oscillating torus, seen in Fig.3, starts filling the region and induces a small accretion process towards the black hole. This process does not create a big shock wave to reduce the rest-mass density of the torus until the snapshot at t = 6070M , as seen in Fig.1. Later, the oscillation amplitude of the torus is getting bigger during the time evolution, and it causes the creation of the spiral shock wave. The spiral wave causes the angular momentum transfer and the matter the falling towards to or away from the black hole. Having a shock wave also causes kinetic energy to convert to thermal energy. The spiral shock waves can be clearly seen in Fig.1 starting from the snapshot t = 7324M .
The spiral density wave traveling with same angular velocity of matter on the torus creates co-rotation radius. Inside this radius, matter travels slowly when it is compared with outside. Therefore, negative angular momentum appears inside the wave which causes the matter to fall towards the black hole.
The position of the maximum rest-mass density in the disk during the evolution is shown in Fig.2 around the black hole for different models. As seen from the figure, the effect of kicked black hole onto the disk dynamic is almost the same for different kick values which are represented with χ in Table 1. However, the positions of the maximum densities make nonlinear oscillation during the evolution. The oscillation might be appeared due to the growth of the gravitational instability (spiral) mode. On the other hand, the maximum rest-mass density of the stable torus was initially at 3.4M . After the perturbation is applied, the position of the maximum rest-mass density is pushed out from the black hole horizon with a significant distance r ∼ 9M around t ∼ 10000M . Transferred angular momentum from the kicked black hole to the torus starts to affect the torus dynamics after a few dynamical periods (∼ 4t orb ). Initial perturbation causes instability on the dynamic of the torus and the smaller the transferred angular momentum, the more instability develops in later time of simulation seen in Fig.2 for the rotating black hole. A higher black hole spin creates a larger centrifugal barrier so that it causes to be less accretions towards to the black hole.
The mass accretion rates in arbitrary unit around the rotating black hole in case of different perturbation velocities are shown in Fig.3. It is computed at inside the torus at r = 3.8M close to the black hole horizon. While the accretion rate dM/dt > 0 represents the perturbed torus' matter falling towards the black hole, the matter moves away from the black hole for dM/dt < 0. Fig.3 has a clear indication of the torus' matter oscillates due to black hole kick. All the three cases in Fig.3 indicate that after significant oscillation phase like sinusoidal functions, the mass accretion rate tends to a fairly constant value. It is found that the amplitude of the perturbation makes some delays in case of having an instability but these perturbations do not strongly responsive to accretion rate in overall.
The torus with a constant specific angular momentum around the black hole is unstable in the non-axisymmetric global mode (Chen et.al. 2021). The non-axisymmetric global mode is produced on the torus due to kicked black hole. In order to the reveal the growth of this global mode m = 1, we compute the power mode amplitude from the density of torus. Power mode is basically computed by finding the Fourier decomposition of the azimuthal distribution of the tours density during the time evolution by using the following equation Donmez (2014) where r in and r out are inner and outer boundaries of the computational domain, respectively. The real and imaginary parts are Re(w m (r)) = 2π 0 ρ(r, φ)cos(mφ)dφ and Im(w m (r)) = 2π 0 ρ(r, φ)sin(mφ)dφ. As seen in Figs.3 and 4, the magnitude of perturbation influences the process of stabilization of the matter and saturation time because the angular momentum of the initially stable torus faces a big influence due to the kicked black hole. It is clearly found that the the density wave seen for m = 1 in exponential growth mode is developed around the rotating black hole. It is shown in Fig.4 that the saturation time is slightly different for different perturbation velocities. The higher the perturbation velocity, the more delayed it is to reach the saturation point. However, the amplitudes of mode m = 1 for different models are almost the same. The growing in instability can happen just by oscillating the black hole-torus system.
The space time diagrams for (φ − t) and (r − t) of the torus logarithmic rest-mass density during the time evolution around the rotating black hole to see how and where the spiral arms are created are shown in Figs.5 and 6, respectively. In order see more detailed dynamics of the accreted torus, whole evolutions of the numerical simulations are separated into three pieced. In the early stages of the simulation for model K09C, the maximum value of density of torus would not be effected too much from the perturbation due to the kicked black hole, seen in Figs.5 and 6. However, the spiral arms are created during the evolution and this would lead to the collimation of the radiation into beams. These types of phenomena could be used to explain why the super Eddington luminosity is observed close to the black hole horizon. The spiral shocks created the hot corona in the inner region of the accretion disk due to frictional and compressional heating. The accretion due to spiral shock varies highly with time and this type of accretion is responsible for the irregular behavior and the quasi-periodic oscillation observed in black hole-torus systems in galaxies and AGN. As it is seen in the middle panel of Fig.5, the oscillation is excited around the maximum of the epicyclic frequency which is 10M in the rotating black hole. Additionally, high energetic interactions inside the spiral arms would scatter emission from the region close to the black hole.
Initial torus with a constant specific angular momentum is rotating around the black hole and perturbed by the black hole itself. These types of perturbations would cause a redistribution of the specific angular momentum of the torus. In order to mimic the time evolution of the redistribution of the specific angular momentum and clarify the effects of spiral shocks, we plot the variation of angular velocity in Fig.7. As it is seen in the top panel of the figure, the instability grows early time of simulation t ∼ 300M , the angular velocity is redistributed and it causes rotating a strong spiral shock which can be seen after t = 1986M . From our numerical simulations it is obvious that the influence of perturbation on angular velocity creates a strong spiral shock which is excited in the disk and propagate through the stable circular orbit. The stronger the spiral shock would lead the matter the accreting toward the black hole and might be very important to explain the spectral properties of the black hole candidates. Bottom panel of Fig.7 has a clear evidence about the creation of a strong shock at certain times. The angular velocity is varying between the smallest and the highest values of the color bar at t ∼ 1500M , t ∼ 2200M , t ∼ 3100M , t ∼ 4700M , etc.

Non-rotating Black Hole
The dynamical evolutions of the rest-mass densities on the equatorial plane after the perturbation show a big difference around the rotating and non-rotating black hole as seen in Figs.1 and  8, respectively. The maximum rest mass density of the accreted torus stays almost bellow its initial value during the time evolution. Having a rotating or non-rotating black hole causes a drastic change on the dynamics of the accreted matter and spiral waves created on the torus.
The kicked non-rotating black hole also leads some excitations of the specific unstable nonaxisymmetric modes. In order to compute the azimuthal number m of the spiral wave mode, the dynamical evolution of the torus is observed for different range of perturbation parameters seen in Eq.5. After the black hole-torus system is perturbed, the new equilibrium stage is almost reached at the saturation time t = 750M , seen in Fig.9. It shows that the kicked black hole enhances the strength of the fastest growing mode m = 1. spiral wave instability driven on the torus can be characterized by the presence of the radial pressure gradient. As it is seen in Fig.4, the matter that  While the black dashed curve shows the angular velocity for the initial torus, the others represent the evolution of the velocity during the time. Bottom panel: The logarithmic angular velocity is represented by a surface whose colors display the variation of the spiral shock along the angular direction. is getting closer to the black hole horizon would lead the stronger spiral density wave mode.

Comparison of Rotating and Non-rotating Black Hole Cases
The rest-mass densities for models at different snapshots are given around the rotating and non-rotating black holes in Figs.1 and 8, respectively. As it is seen from these two cases, the effect of black hole spin onto the torus as well as the morphology of torus after perturbation are clearly seen. As it is also seen in Fig.10, the matter is pushed away from the center of the black hole due to the oscillating black hole being connected to the black hole horizon with a spiral wave, even in the last snapshot of simulation for the rotating black hole this can be seen. Similarly, we can also clearly see the structure of the spiral arms close to the non-rotating black hole, shown in Fig.8. We observed significant differences in the morphology of the disk when the black hole is rotating even though we used the same initial perturbation parameters.
Overall, the mass accretion rates computed at the inner boundaries around the non-rotating and the rotating black holes show similar behavior during the evolution seen in Figs.10 and 11, but our numerical simulations of the perturbed torus due to the kicked black hole indicate that transition from unstable state to steady-state shows clear evidence of dependencies on the black hole spin. We find that the accretion rates are consistently bigger for the larger black hole spin. The reason for this is that the maximum density of the initial stable torus around the rotating black hole is closer to the black hole horizon. Kicked black hole makes a big impact on the torus dynamics whenever the black hole is kicked. It is found that a higher spin produces a higher accretion rate, which implies the higher critical temperature, during the evolution. The increased in the mass accretion rate causes the emitting radiation in the high-energy. Therefore, the torus-black hole system may be a source of continues high energetic emissions in several channels, possibly in association with GRBs (van Putten 2002).
The saturation time of the instability around the rotating black hole is t = 1350, but it is almost half t = 750 around the non-rotating black hole. This means that the saturation times and growth of the instability strongly depend on the black hole spin. After the saturations, the torus still keeps losing the matter either towards or away from the black hole due to redistribution of the angular momentum and we still witness the drastic changes in the torus dynamics during the evolution. Additionally, as it is seen in Figs. 4 and 9, the amplitude of the power of the m = 1 mode around the rotating black hole is larger than that for the non-rotating black hole. It is a clear evidence for dependency of the mode amplitude on the black hole spin.
Studying the angular momentum transfer around the black hole would tell us how the accretion happens through the spiral shock wave. In order to show the effect of the black hole spin Here, we will find out how the perturbed torus transports the angular momentum along the equatorial plane; radially inward or outward and its dependencies to the black hole spin parameter.
The perturbed torus around the black hole creates a spiral shock wave around the rotating and non-rotating black holes. The angular momentum of the rotating gas can play an important role in the moving of gas away from the black hole or towards it. The angular momentum of the initial torus is constant in the begging of simulation. After perturbation is felt by the torus, it is clearly seen in Fig.12 that the angular momentum flux varies depending on the black hole rotation parameter. The strength of oscillating angular momentum flux is at least 100 times bigger in case of torus around the rotating black hole. The higher the black hole spin parameter leads the bigger oscillation in the angular momentum transfer. The angular momentum flux almost oscillates around zero value for the rotating black hole. The spikes of the angular momentum transport has a strong correlation with presence of spiral shock, seen in Fig.7.
In addition to power mode evolutions on the non-axisymmetric modes around the rotating and non-rotating black hole given in Figs.4 and 9, we explore more detail information about the local and non-axisymmetric instability using the rest-mass density. The rest-mass density is one of the perturbed physical variable of torus when the black hole oscillates inside the galaxy. The time evolutions of the maximum value of the rest-mass density in logarithmic scale around the non-rotating and rotating black holes are given in Fig.13. The non-axisymmetric perturbations display a significant growth during certain time interval ( until t ∼ 10000M ) in both cases. It almost oscillates around some equilibrium point after this time. The amplitude, however, strongly depends on the black hole spin parameter.

Discussion and Conclusion
We have studied the dynamical evolution of the torus around the black hole that undergoes perturbation due to the kick, presumably it is the gravitational radiation from the mergers of the black holes. There is a number of black hole-accretion disk systems that might be considered as candidates for the perturbed torus due to the kicked black hole. The results found in our numerical  simulations are given in geometrized units; therefore, they might be used to explain many of the perturbed the kicked black hole-torus systems.
In order to understand the dynamical feature of the torus with respect to the perturbation amplitude produced by the kicked black hole, we have varied the parameter χ, which is a free parameter to control the perturbation on the torus radial velocity, and it is found that different values of χ produce almost the same behavior for given black hole-torus system. Only some time delays when the spiral density wave instability was reached the saturation point and types of shock waves created during the time evolution were observed. It is clearly found that the instability seen for m = 1 in the exponential growth mode is developed around the rotating and non-rotating black holes. The non-axisymmetric growing mode m = 1 causes the formation of the spiral shock wave appeared in the rotating matter around the black holes (Mewes et. al. 2016).
The mass accretion rates, which appear around rotating black hole almost 100 times bigger than the one around the non-rotating black hole, show that the accretion associated with the saturation of spiral density growth mode occurs slightly earlier in case of the rotating black holes. Additionally, the location of the maximum density of the perturbed torus around the rotating black hole gets closer with the non-rotating case after they reach the saturation point, event though they are initially operated with a distance of 3.40M .
We have observed the instability in our models which have developed due to the kick that the center of black hole-torus system receives. The kicked black hole triggers the radial nonaxisymmetric oscillation on the disk. And these oscillations on the equilibrium stage of the torus lead to the spiral density wave. When the growing spiral azimuthal mode occurs just before or after reaching of the saturation point could trigger it not only in the torus, but also in the black hole. It would cause either an increase or an decrease in the black hole kick velocity (Mewes 2016).
The rest-mass density oscillation, the growth of instability and the resulting spiral shock waves that we have found in our numerical simulations during the time evolution could play an important role in explaining the vigorous X-rays phenomena observed from the galactical nuclei and the quasars.