Effects of resistivity on standing shocks in low angular momentum flows around black holes
aa r X i v : . [ a s t r o - ph . H E ] F e b Research in Astron. Astrophys. Vol.0 (20xx) No.0, 000–000 (L A TEX: ms2020-0357.tex; printed on February 12, 2021; 1:17) R esearchin A stronomyand A strophysicsReceived 20xx month day; accepted 20xx month day Effects of resistivity on standing shocks in low angular momentumflows around black holes
Chandra B. Singh , Toru Okuda and Ramiz Aktar South-Western Institute for Astronomy Research, Yunnan University, University Town, Chenggong,Kunming 650500, People’s Republic of China; [email protected]. Hakodate Campus, Hokkaido University of Education, Hachiman-Cho 1-2, Hakodate, Hokkaido040-8567, Japan Department of Astronomy, Xiamen University, Xiamen, Fujian 361005, People’s Republic of China
Abstract
We study two dimensional low angular momentum flow around the black holeusing the resistive magnetohydrodynamic module of PLUTO code. Simulations have beenperformed for the flows with parameters of specific angular momentum, specific energy,and magnetic field which may be expected for the flow around Sgr A*. For flows withlower resistivity η = 10 − and . , the luminosity and the shock location on the equa-tor vary quasi-periodically. The power density spectra of luminosity variation show thepeak frequencies which correspond to the periods of 5 × , 1.4 × , and 5 × sec-onds, respectively. These quasi-periodic oscillations (QPOs) occur due to the interactionbetween the outer oscillating standing shock and the inner weak shocks occurring at theinnermost hot blob. While for cases with higher resistivity η = 0 . and 1.0, the high resis-tivity considerably suppresses the magnetic activity such as the MHD turbulence and theflows tend to be steady and symmetric to the equator. The steady standing shock is formedmore outward compared with the hydrodynamical flow. The low angular momentum flowmodel with the above flow parameters and with low resistivity has a possibility for theexplanation of the long-term flares with ∼ one per day and ∼ Key words: accretion, accretion disks — magnetohydrodynamics (MHD) — meth-ods:numerical — shock waves — Galaxy: center
Black hole accretion is the most efficient process which can address the issue of power generated inthe neighbourhood of a black hole. Historically, the study of black hole accretion has been based ontwo extreme cases of accretion process: radiatively inefficient flow called Bondi flow (Bondi (1952);Michel (1972)) and radiatively efficient one called Keplerian disk (Shakura & Sunyaev (1973); Novikov& Thorne (1973)). Both suffer from certain limitations. Spherical Bondi flow with zero angular mo-mentum is quite fast and cannot explain the high luminosities associated with observational signaturesaround the black hole. However, in reality, accretion flow is supposed to have some amount of angularmomentum associated with it. On the other hand, cold, thin, Keplerian disk cannot explain the issue ofchange of spectral states and associated temporal variabilities and it is not applicable for the close regionaround the black hole as pressure gradient and advective radial velocity terms are ignored.
Singh, Okuda & Aktar
Accretion flow onto the black hole is supposed to be supersonic at the event horizon and subsonic ata large distance as the accretion flow approaches the speed of light at the horizon with sound speed beingof lesser value. So the flow with angular momentum must pass through at least one sonic point beforeplunging onto the black hole and should be advective (Liang & Thompson (1980)). In case of accretiononto a star, even a small angular momentum will stop the matter fall onto its surface because of theinfinite potential barrier associated with the Newtonian potential. Whereas in the case of a black hole,gravity always wins over centrifugal force because of higher-order terms (Chakrabarti (1993)). Notonly that, for given values of specific energy and specific angular momentum of accretion flow aroundthe black hole, multiple sonic points may also exist with the possibility of standing shocks (Fukue(1987); Chakrabarti (1989)). There have been several interesting works which explored the solutionswith standing shocks in pseudo-Newtonian potential (Paczy´nsky & Wiita (1980)) taking into accountvarious prescriptions for alpha parameter (Chakrabarti (1996); Becker, Das & Le (2008); Kumar &Chattopadhyay (2013)). General relativistic solutions for inviscid (Das et al. (2015)) and viscous disks(Chattopadhyay & Kumar (2016)) with standing shocks have also been reported. Even in the presenceof magnetic field, formation of standing shocks in accretion flows have been explored (Takahashi et al.(2006); Fukumura et al. (2007)).In recent time the model which has wide recognition is the advection dominated accretion flow(ADAF) (Narayan & Yi (1994); Narayan, Kato & Honma (1997)) solution which takes care of innerboundary condition around the black hole however, has only one sonic point close to the black hole. Itshould be noted that advective flow with multiple sonic points may not necessarily be ADAF-type espe-cially when standing shock exists in the accretion flow (Chakrabarti (1996)). Overall, ADAF solutionsoccupy a small region of parameter space for given specific energy and specific angular momentum (Lu,Gu, Yuan (1999); Kumar & Chattopadhyay (2013, 2014)). The need of a sub-Keplerian componentwas presented in addition to Keplerian one, the sub-Keplerian component can undergo shock transi-tion and form a hot, puffed up region like corona (Chakrabarti & Titarchuk (1995)). The properties ofpost-shock region formed from the natural course of flow dynamics can address issues like state transi-tions (Mandal & Chakrabarti (2010)), origin of hard power-tail and low frequency QPOs (Chakrabarti,Mondal & Debnath (2015)) and also the origin of outflows (Das et al. (2001); Singh & Chakrabarti(2011); Aktar, Das & Nandi (2015)).In the last 25 years, there have been a significant amount of simulation works dedicated to ex-plore the formation of standing shock in low angular momentum sub-Keplerian advective flows aroundblack holes. Using smoothed particle hydrodynamics (SPH) simulations, stable standing shocks wereshown to form in one-dimensional (Chakrabarti & Molteni (1993)) and two-dimensional setups(Molteni, Lanzafame & Chakrabarti (1994)) as predicted by semi-analytical solutions of inviscid flows(Chakrabarti (1989)). For the first time, the dependence of standing shock stability on values of viscos-ity parameters was shown by SPH simulations as well (Chakrabarti & Molteni (1995)). The origin ofoutflows from the post-shock region in accretion disks were shown in simulations using SPH (Molteni,Ryu & Chakrabarti (1996)), Eulerian total variation diminishing (TVD) (Ryu, Chakrabarti & Molteni(1997); Okuda, Teresi & Molteni (2007)) and Lagrangian TVD (Lee et al. (2016)). In the presence ofcooling, the post-shock region may oscillate as the cooling time scale becomes comparable to free-falltime scale and can be responsible for quasi-periodic oscillations(QPOs) in case of stellar mass as well assupermassive black holes (Molteni, Sponholz & Chakrabarti (1996); Okuda, Teresi & Molteni (2007)).Besides the case of inviscid flow, viscosity can also induce shock oscillations and give rise to QPOs(Lanzafame, Molteni & Chakrabarti (1998); Chakrabarti, Acharrya & Molteni (2004); Lanzafame et al.(2008); Lee, Ryu & Chattopadhyay (2011); Das et al. (2014); Lee et al. (2016)). There have been someworks regarding stability or instability of the shock and shocks seem to be stable against axisymmetric(Nakayama (1992, 1994); Nobuta & Hanawa (1994); Le et al. (2016)) as well as non-axisymmetricperturbations (Molteni, Toth & Kunetsov (1999); Gu & Foglizzo (2003); Gu & Lu (2006)). Recentlyit has established through numerical simulations that advective flow can be segregated into two com-ponents, Keplerian as well as sub-Keplerian, in the presence of viscous heating and cooling processes(Giri & Chakrabarti (2013); Giri, Garain & Chakrabarti (2015); Roy & Chakrabarti (2017)). All theabove-mentioned simulation works addressed the accretion flow behaviour around a non-rotating black esistive advective flows with standing shocks 3 hole using pseudo-Newtonian potential. Recently general relativistic high-resolution shock-capturingsimulation code was used to study the scenario in the Schwarzschild (Kim et al. (2017)) and Kerr (Kimet al. (2019)) space-time which further established the formation of standing shock in hydrodynamicflow around non-rotating as well as rotating black hole in full general relativistic treatment. However, tillnow there has been only one work taking into account different magnitudes of magnetic field strength insuch flows in the presence of standing shocks (Okuda et al. (2019)). The long term evolution propertieswere investigated and long term flares in connection with Sgr A* could be explained.Simulation works dealing with advective flows usually take into account two types of setup : torusequilibrium solution (e.g., (Stone & Pringle (2001); McKinney, Gammie (2002)) and Bondi flowalong with arbitrary choice of specific angular momentum (Proga & Begelman (2003)). Our studyinvolves a third and different kind of set up where we take initial conditions based on exact solutions ofhydrodynamic equations (Chakrabarti (1989)). We are dealing with a big gap in parameter space whichlie between the regime of high angular momentum torus and zero angular momentum Bondi flow. Tomake our study simpler, we are dealing with inviscid flows having constant and small specific angularmomentum value which are lower than that of Keplerian value of the specific angular momentum for aninner-most stable orbit. Such low angular momentum flows are likely to be present in binary systemsaccreting winds from companion beside Roche lobe overflow as well as active galactic nuclei wherewinds from stellar clusters collide and lose angular momentum before getting accreted onto central blackholes (Chakrabarti & Titarchuk (1995); Smith et al. (2001, 2002); Moscibrodzka et al. (2006)). Theobjective of our work is to study the effects of resistivity with the varying magnitude on the formationand stability of standing shocks in low angular momentum accretion flows around the black holes whichhave not been explored before.Section 2 shows details of 1.5 dimensional (1.5D) theoretical solutions which have been used forthe simulation set up. In section 3, basic equations solved by simulation code are presented. Besides, thedetails of computational domain, initial, and boundary conditions are described in section 3. Section 4contains details of numerical results followed by section 5 where we present a summary and discussionof our work.
We consider a semi-analytical approach of solving the standard conservation equations under hydrody-namic (HD) framework. The calculations are done in cylindrical co-ordinates with co-ordinates R and z . Axi-symmetry is assumed for the angular φ co-ordinate. For simplicity, we further assume that theflow velocity along the vertical direction is zero and therefore only integrate along the radial co-ordinateassuming vertically averaged dynamical quantities.We define the scale radius as Schwarzschild radius r g = GM/c , with M being the mass of thecentral compact object, G as the gravitational constant and c is the speed of light. The matter that isaccreted onto the central compact object has radial velocity given by u R , specific angular momentum L ,and total specific energy E . As the semi-analytical calculations are carried out using non-dimensionalquantities, we define the following - r = Rr g ; h = zr g ; v R = u R c ; λ = Lr g c ; ǫ = E c (1)For studying ideal, inviscid flow onto a compact object, we deal with mass conservation equation, ˙ M = 4 πρv R rh, (2)where ρ is the density and h being the half-thickness of flow. Energy conservation gives us a relation ofspecific energy of the flow or Bernoulli constant, ǫ = v c s Γ − λ r + Φ . (3) Singh, Okuda & Aktar
Fig. 1: Mach number versus radial distance from semi-analytical estimates for flow parameters, specificenergy, ǫ = 1.98 × − and specific angular momentum, λ = 1 . , with Γ = 1.6 (adopted from Okudaet al. (2019)).Here c s is adiabatic sound speed in units of c and Γ is the adiabatic index. c s = q Γ pρc . p is the thermalpressure and Φ is the non-dimensional gravitational potential given by − / x − for non-rotatingblack hole (Paczy´nsky & Wiita (1980)) where x = r sp /r g and r sp being the spherical radius.Using vertical equilibrium condition, we evaluate the radial dependence for hh = c s √ x ( x − (4)For given values of ǫ and λ , we solve equations (2) and (3) and look for transonic conditions.Differentiating the equations (2) and (3), we obtain dv R dr [ v R − c s (Γ + 1) v R ] = 2 c s Γ + 1 dlnfdr − dGdr , (5)where G = λ / r − / r − and f = 2 r / ( r − (Chakrabarti (1989)). At critical points, the vanishing of left-hand side gives radial velocity, ( v R ) crit = r
2Γ + 1 ( c s ) crit , (6)and the vanishing of right hand side gives sound speed, ( c s ) crit = (Γ + 1)( r crit − r crit ( λ K − λ )(5 r crit − . (7)The subscript crit and K represent quantities at the critical points and Keplerian orbits respectively.In the case we obtain multiple critical points, we also check whether shock conditions are satisfied ornot along the accretion flow. If shock conditions are satisfied, then there is a possibility of standingaxisymmetric shock otherwise not. The shock location can be determined using an invariant quantity, C , across the shock which is given by esistive advective flows with standing shocks 5 C = [ M + (3Γ −
1) + (2 / M + )] − M = [ M − (3Γ −
1) + (2 / M − )] − M − . (8)Here, M = v R /c s is the Mach number of the accretion flow. The subscripts − and + represent quantitiesin the pre-shock and post-shock region respectively. Further details of the semi-analytical approach canbe found in (Chakrabarti (1989)).Fig. 1 shows the variation of Mach number, M , of flow with radial distance from black hole ob-tained from exact theoretical solution solving conservation equations. The transonic flow passes throughthe outer critical point “a” and continues its journey towards the black hole. The flow chooses to un-dergo shock transition along “bc”, becomes subsonic then again accelerates towards the black hole andpasses through inner critical point “d” to become supersonic before entering the black hole horizon. Theshocked flow is preferrable in nature as the entropy generation is relatively higher compared to no shockflow. The numerical setup for the present work uses grid-based, finite volume computational fluid dynamicscode, PLUTO (Mignone et al. (2007, 2012)). Numerical simulations are carried out by solving theequations of classical resistive magnetohydrodynamics (MHD) in the conservative form: ∂ρ∂t + ∇ · ( ρ v ) = 0 , (9) ∂ ( ρ v ) ∂t + ∇ · [ ρ vv − BB ] + ∇ p t = − ρ ∇ Φ , (10) ∂E∂t + ∇ · [( E + p t ) v − ( v · B ) B + η ( ∇ × B ) × B ] = − ρ v · ∇ Φ , (11) ∂ B ∂t − ∇ × ( v × B − η ∇ × B ) = 0 . (12)Here, p t being the total pressure with contribution from thermal pressure, p , and magnetic pressure, B / . E is the total energy density given by E = p Γ − ρ v + B ) . (13) η is the resistivity for which range of values have been chosen, − , . , . and . Vector potential A is prescribed to generate the magnetic field B as B = ∇ × A . Following Proga & Begelman (2003), thecomponents of A are as follows A R = 0 , A φ = A zr sp R and A z = 0 . Here, A = sign ( z )( πp out β out ) / R out and β out = 8 πp out /B out , subscript ”out” denotes parameters at the outer boundary of the computationaldomain, R out . Following Okuda et al. (2019), we take a typical value of 5000 for β out . First, we have studied advective flows onto black holes in axisymmetric 2D cylindrical geometry ( R , z ) in HD framework. The theoretical solutions given in section 2 provide initial conditions of primitivevariables, radial, and azimuthal velocity components, density, and pressure for 2D HD set up. Once theHD flow with the standing shock achieves a steady state, we use the solutions as the initial conditionsfor the magnetized flow with resistivity and let the simulation evolve further. The computational domainis ≤ R ≤ R g and − R g ≤ z ≤ R g with the resolution of × cells (for details, seeOkuda et al. (2019)). Though we performed some simulation runs at half resolution, × , and Singh, Okuda & Aktar
R/Rg l og l ogT v T v R ρ ρ R Fig. 2: Flow parameters on the equator, namely density ( ρ in g cm − ), radial velocity ( v R ) and tempera-ture (T), for final state of HD simulation run. The standing shock is at . R g (adapted from Okuda etal. (2019)).double resolution, × , the results remain unchanged.In both HD and MHD runs, the same boundary conditions are imposed. At the outer radial boundary, R out = 200 R g , there are two domains: the disk region where the matter is injected and the atmosphereabove the disk region. The flow parameters given by 1.5D theoretical solutions are provided in theregion − h out ≤ z ≤ h out where h out is the vertical equilibrium height at R out . For the atmosphereregion, the matter is allowed to leave the domain but not enter. The axisymmetric boundary conditionis implemented at the inner radial boundary. At R = 2 R g , the absorbing condition is imposed in thecomputational domain. In the vertical direction, z = ± R g , standard outflow boundary conditions areimposed. In the case of MHD run, the constant magnetic field is imposed on the outer radial boundary.Fig. 2 shows profiles of density ( ρ in g cm − ), radial velocity ( v R ) and temperature(T) of the HD flow.The standing shock location from the simulation run is at ∼ R g which is significantly different fromthe predicted location from the theoretical solution, i.e. ∼ R g . This is due to the assumption of thevertical hydrostatic equilibrium used in the 1.5D transonic solutions, which is valid as far as the diskthickness h is sufficiently small compared with the radius r (that is, h/r ≪ h/r ∼ To get the characteristic features of the flow, we examine the time evolution of shock location R s on theequator and total luminosity L of the flow. The luminosity L is calculated as follows, assuming that thegas is optically thin L = Z q ff dV, (14) esistive advective flows with standing shocks 7 - . - . - . - . - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R/Rg z / R g log Qr η = 10 −6 - . . . . . . . . . . . . . . . . . . . . . . . . . . . R/Rg z / R g log Qr η=1.0 Fig. 3: 2D contours of MRI-criterion Q r for cases of η = 10 − and 1.0 at times t = × and . × seconds, respectively. For both cases, Q r ≫ q ff represents the free-free emission rate per unit volume and integration is performed over all thewhole computational domain. In order to have some estimate of how much matter is lost as an outflowfrom the system, mass outflow rate in the z-direction is given by, ˙ M out = Z R out πρ ( R, z out ) v z ( R, z out ) RdR − Z R out πρ ( R, − z out ) v z ( R, − z out ) RdR, (15)where v z is the vertical velocity.In the present study, the time variability of the flow will correlate with the magnetorotational insta-bility (MRI). We check whether the flow is subject to the MRI and whether we can resolve the fastestgrowing MRI mode or not. The stringent diagnostics of spatial resolution for the MRI instability hasbeen examined in 3D magnetized flow. Therefore, its application to our 2D magnetized flow may belimited. The critical wavelength of the instability mode is given by λ c = 2 πv A / √ , where v A and Ω are the Alfven velocity and the angular velocity (Hawley & Balbus (1991); Balbus & Hawley (1998)).A criterion value Q x of the MRI resolution is defined by Q x = λ c ∆ x , (16)where ∆ x is the mesh sizes ∆ R and ∆ z in the radial and vertical directions, respectively. When Q x ≫
1, the flow is unstable against the MRI instability, otherwise, the flow is stable. Fig. 3 shows 2D contoursof radial MRI-criterion Q r for cases of η = 10 − and 1.0 at times t = × and . × seconds,respectively. The analyses show Q r ≫ Fig. 4 shows how luminosity L and shock location R s vary with time for different levels of resistivityin the flow. For lower values of resistivity, η = 10 − and . , there are features of irregular oscillationin the luminosity L and the standing shock location R s . As the shock moves towards the black hole,there is an increase in luminosity while the luminosity decreases when the shock recedes away. Theshock and the luminosity oscillate irregularly with time scales of ∼ − s, and the luminosityvaries maximumly by a factor of ten around the average L is ∼ . × erg s − . On the other hand, Singh, Okuda & Aktar
Fig. 4: Variation of Luminosity (L) and shock location ( R s ) with time for resistive MHD flow withdifferent values of resistivity, η = − , 0.01, 0.1 and 1 (in clockwise direction).for relatively high resistivity, . and . , the oscillatory nature disappears and L and R s show smallmodulations or almost steady value at later times. The highly resistive flows behave qualitatively similarto that of HD flow.The mass inflow ˙ M edge and outflow ˙ M out rates are presented in Fig. 5 correspondings to Fig. 4.Similar to our previous work Okuda et al. (2019), it has been established that there is a correlationbetween L and ˙ M edge and between R s and ˙ M out . While the variation of L seems to be opposite inbehaviour compared to R s . That means when the post-shock region shrinks, the emission increases,and vice versa. Since the low angular momentum flows are very advective, most of the input gas ˙ M input ( ∼ × g s − ) falls into the event horizon and ˙ M edge is comparable to ˙ M input in all cases. However,the mass outflow rate ˙ M out in low resistivity case is considerably high as a few tens of percent of theinput accretion rate but in the high resistive case with η = 0.1 and 1.0, ˙ M out amounts to ∼
10 percent.Such mass outflow rates are very high compared with the mass outflow rate found in the usual accretionflow. It should be noted that the very high mass outflow rate in the low resistivity case may be correlatedto the MRI turbulence, that is, the MRI turbulence plays important roles not only in the outward transferof the angular momentum but also in outward mass transfer. esistive advective flows with standing shocks 9
Fig. 5: Mass inflow ( ˙ M edge ) as well as outflow ( ˙ M out ) rate evolving with time for resistive MHD flowwith different values of resistivity, η = − , 0.01, 0.1 and 1 (in clockwise direction). The resistivity has dissipative and diffusive characters in the magnetic field through the current den-sity, similar to the viscosity in hydrodynamical flow and we expect the higher resistivity to suppressthe magnetic activity like magnetic turbulence. We examine the effects of resistivity through the timeevolution of the magnetized flows with η = 10 − to 1.0. In the case with the lowest resistivity − ,after a transient initial time evolution, the magnetic field is amplified rapidly by the MRI and the MHDturbulence develops near the equatorial plane. Fig. 6 shows radial profiles of normalized Reynolds stress α gas and normalized Maxwell stress α mag for resistive MHD flow with different resistivity, η = − ,0.01, 0.1 and 1 (in clockwise direction). These values are space-averaged (between − R g and R g inthe z-direction) and the time averaged over the last duration time. Here we see that the maxwell stress Σ mag is larger by a factor of a few to ten than the Reynolds stress Σ gas in cases of lower η = 10 − and0.01 over the most region, while in the higher η the Reynolds stress mostly dominates over the Maxwellstress. From this, we confirm that the higher resistivity suppresses the Maxwell stress and then MHDturbulence, that is, hydrodynamical mode dominates over magnetohydrodynamical mode. As the result,in the case with the highest resistivity, the flow is dominated by the hydrodynamical quantities at theouter radial boundary which are symmetric to the equator and the flow achieves a steady and symmetricstate. Fig. 7 denotes radial profiles of the gas pressure and the magnetic pressure for MHD flow with Fig. 6: Radial profiles of normalized Reynolds stress α gas and normalized Maxwell stress α mag forresistive MHD flow with different resistivity, η = − , 0.01, 0.1 and 1 (in clockwise direction). Thesevalues are space-averaged (between − R g and R g in z-direction) and time averaged (between . × and . × seconds).resistivity, η = − , 0.01, 0.1 and 1 (in clockwise direction). In all cases with different resistivity, thegas pressure dominates the magnetic pressure and the pressure distributions are not so different eachother.Fig. 8 shows 2D density contours and velocity vectors at the later evolution of the flow with η = 10 − and 1.0 at times t = 7 × and 8.7 × seconds respectively. Here, the location ofstanding shock is distinguished as the thick black contour lines, and the velocity vectors are taken to bean arbitrary unit. In the low resistivity case, the density contours are asymmetric to the equator and tur-bulent motions are observed in the shocked region. While in the high resistivity case variables becomesymmetric to the equator and no turbulent motion is observed. The flow features seem to return to theinitial hydrodynamical steady-state but with a bit larger shock location R s ∼ R g than R s ∼ R g in Fig. 2, because the magnetic pressure contributes to the pressure balance to some extent in the shocklocation. The present results for cases with low resistivity of η = 10 − and 0.01 are very similar to those for theprevious magnetized flow without resistivity (Okuda et al. (2019)). Adopting the same parameters ofthe flow and magnetic field as the present study, they found that the centrifugally supported shock movesback and forth between 60 R g ≤ R ≤ R g and that another inner weak shock appears irregularlywith rapid variations due to the interaction of the expanding high magnetic blob with the accreting matterbelow the outer shock. The process repeats irregularly with an approximate time-scale of (4 – 5) × s ( ∼ ∼ . × s (25hrs). In this respect, we also analyzed the time variability of the resistive magnetized flows. Fig. 9 showthe power density spectra of luminosity for different values of resistivity. For η = − and . , thepeak (fundamental) frequency is estimated roughly to be at 2 × − along with two weak signatures(harmonics) at 7 × − and 2 × − Hz. These correspond to the periods of 5 × s (5.8 days), 1.4 esistive advective flows with standing shocks 11 Fig. 7: Radial profiles of gas pressure and magnetic pressure which are space-averaged (between − R g and R g in z-direction) and time averaged (between . × and . × seconds) for resistive MHDflow with different resistivity, η = − , 0.01, 0.1 and 1 (in clockwise direction). × s (1.6 days), and 5 × s (0.6 day) , respectively and are comparable to two QPOs periods ∼ days and ∼ day found in the non-resistive magnetized flow. Therefore the QPO peak frequencies canbe associated with periods of ∼ ∼ η = . and . , there is no clear peakfrequency. The average mass outflow rate ∼ − M ⊙ yr − and mass inflow rate ∼ × − M ⊙ yr − obtained in small η cases show a roughly good correspondence with the Chandra observations (Wang etal. (2013)) which suggest the presence of a high outflow rate that nearly balances the inflow rate. We studied the effect of resistivity on standing shock in the magnetized flow around a black hole. Theflow parameters of specific energy, ǫ = 1.98 × − and specific angular momentum, λ = 1 . , with Γ = 1.6 have been considered to address the flow behaviour around Sgr A*. For flows with lower resis-tivity η = 10 − and . , the luminosity and the shock location on the equator vary quasi-periodically.These quasi-periodic oscillations are attributed to the interactive result between the outer oscillatingstanding shock and the inner weak shocks occurring at the innermost hot blob. The luminosity variesmaximumly by a factor of ten around the average L ∼ . × erg s − . The mass outflow rate isvery large as a few tens of percent of the input accretion rate. The MHD turbulence seems to play im-portant roles in the outward transport of not only angular momentum but also accreting gas. The powerdensity spectra of luminosity variation show the peak frequencies which correspond to the periods of 5 × s (5.8 days), 1.4 × s (1.6 days) and 5 × s (0.6 day) , respectively. While for cases with higher R/Rg z / R g -16.4-17-17.6-18.2-18.8-19.4-20 Time= 7.0E6 sec = 10 log ρ - 6 η R/Rg z / R g -16.2-17-17.8-18.6-19.4-20.2-21 Time= 8.7E6 sec = 1.0 log ρη Fig. 8: 2D density contours and velocity vectors of flows with η = 10 − and 1.0 at times t = 7 × and 8.7 × seconds, respectively. In the former, variables of density and temperature are asymmetricto the equator and turbulent motions are observed within the post-shock region but in the latter casethe flow is almost symmetric to the equator, and no turbulent motion appears. The shock locations aredenoted by thick black contour lines.resistivity η = 0 . and 1.0 the flow becomes steady and symmetric to the equator. Variable features ofthe luminosity disappear here and the steady standing shock is formed more outward compared with thehydrodynamical flow. The mass outflow rate is also high as ∼ a few tens % of the input gas. The highresistivity considerably suppresses the magnetic activity such as the MHD turbulence and tends to formthe magnetized flow to be stable and symmetric to the equator. The low angular momentum magnetizedflow model with low resistivity has possibility for the explanations of the high mass outflow rate ∼ as10 % of the Bondi accretion rate ∼ × − M ⊙ yr − as suggested by Chandra observations (Wang etal. (2013)) and of the long-term flares with ∼ one per day and ∼ Acknowledgements
CBS is supported by the National Natural Science Foundation of China undergrant no. 12073021. RA acknowledges support from National Science Foundation of China undergrant No. 11373002, and Natural Science Foundation of Fujian Province of China under grant No.2018J01007.
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