Effects of Anisotropic Viscosity on the Evolution of Active Galactic Nuclei Bubbles in Galaxy Clusters
DDraft version September 17, 2019
Preprint typeset using LATEX style emulateapj gaspari-mod v. feb18
EFFECTS OF ANISOTROPIC VISCOSITY ON THE EVOLUTION OF ACTIVEGALACTIC NUCLEI BUBBLES IN GALAXY CLUSTERS
Matthew Kingsland , H.-Y. Karen Yang , Christopher S. Reynolds , John A. Zuhone Abstract
The interaction between jets from active galactic nuclei (AGNs) and the intracluster medium (ICM)provides key constraints on the feeding and feedback of supermassive black holes. Much understandingabout AGN feedback is gained from purely hydrodynamic models; however, whether such an approx-imation is adequate for the magnetized, weakly collisional ICM needs to be critically examined. Forexample, AGN-blown bubbles in hydrodynamic simulations are easily disrupted by fluid instabilities,making it difficult to explain the coherence of observed bubbles such as the northwest ghost bubble inPerseus. In order to investigate whether magnetic tension and viscosity in realistic conditions couldpreserve the bubble integrity, we performed the first Braginskii-magnetohydrodynamic simulations ofjet-inflated bubbles in a medium with tangled magnetic field. We find that magnetic tension aloneis insufficient to prevent bubble deformation due to large velocity shear at early stage of the evolu-tion. Although unsuppressed anisotropic viscosity in tangled magnetic field can have similar effectsas isotropic viscosity, when the pressure anisotropy is bounded by microscopic plasma instabilities,the level of viscosity is substantially limited, thereby failing to prevent bubble deformation as in theinviscid case. Our results suggest that Braginskii viscosity is unlikely to be the primary mechanismfor suppressing the fluid instabilities for AGN bubbles, and it remains a challenging task to repro-duce smooth and coherent bubbles as observed. Because the dynamical influence and heating ofthe ICM critically depend on the bubble morphology, our study highlights the fundamental role of“microphysics” on the macroscopic properties of AGN feedback processes.
Keywords: plasmas — galaxies: active — galaxies: clusters: intracluster medium — magnetohydro-dynamics (MHD) — methods: numerical INTRODUCTION
Feeding and feedback of supermassive black holes(SMBH) are crucial processes determining the evolutionof galaxies and galaxy clusters. Despite being the mostpromising mechanism for solving the “cooling-flow prob-lem” in cool-core (CC) clusters (McNamara & Nulsen2012), the details of active-galactic-nucleus (AGN) feed-back to the intracluster medium (ICM) remain highlydebated.A lot of our understanding about AGN feedback isgained by purely hydrodynamic simulations, from sim-ulations of bubble-ICM interaction (e.g., Churazov et al.2001; Omma et al. 2004; Guo et al. 2018), to simula-tions of self-regulated AGN feedback (e.g., Sijacki et al.2007; Yang et al. 2012; Li & Bryan 2014; Prasad et al.2017). These advances have provided valuable insightsinto the fundamental processes of chaotic cold accretion(e.g., Pizzolato & Soker 2005; Gaspari et al. 2013) andthermalization and distribution of the jet energy (e.g.,Yang & Reynolds 2016b; Li et al. 2017; Ruszkowski et al.2017; Martizzi et al. 2019).Despite the substantial progress, whether ideal hydro-dynamic models are good representations of the ICMis still an open question. For instance, hydrodynamicbubbles tend to be elongated, whereas the “fat" bubblesnear the center of Perseus may be inflated by cosmic-
Email: [email protected] Department of Astronomy, University of Maryland, CollegePark, MD, USA Institute of Astronomy, University of Cambridge, Cam-bridge, UK Harvard-Smithsonian Center for Astrophysics, Cambridge,MA, USA ray (CR) dominated jets (Guo & Mathews 2011; Yanget al. 2019). The morphology of the “ghost" bubble inthe northwest (NW) region of Perseus is also nontriv-ial to reproduce by purely hydrodynamic models due tothe shorter timescales of hydrodynamic instabilities com-pared to the inferred age of the bubble ( ∼ − Myr;Dunn et al. 2005).To preserve the bubble coherence, some additionalphysical mechanisms were invoked, including magneticfield (e.g., Robinson et al. 2004; Ruszkowski et al. 2008)and viscosity (e.g., Reynolds et al. 2005; Guo 2015). Al-though bubbles in their simulations could be stabilized,these studies suffer from a few simplified assumptions,such as studying initially static cavities instead of jet-inflated bubbles, or using isotropic viscosity while theviscosity is expected to be anisotropic in the weakly col-lisional, magnetized ICM. Dong & Stone (2009) improvedprevious works by considering anisotropic/Braginskii vis-cosity along magnetic field lines; however, the survival ofthe bubbles depends on the simplistic field topology as-sumed.To this end, we perform three-dimensional (3D) mag-netohydrodynamic (MHD) simulations of AGN jet-inflated bubbles and investigate the bubble evolutionunder the influence of anisotropic viscosity and tan-gled magnetic field. Specifically, we compare resultsfor four cases: inviscid, isotropic viscosity, unsup-pressed anisotropic viscosity, and anisotropic viscositysuppressed by plasma instabilities on microscopic scales.The last case is motivated by the recent findings thatanisotropic viscosity in weakly collisional plasmas, whichoriginates from pressure anisotropies (see § 2), can begreatly suppressed due to firehose/mirror instabilities a r X i v : . [ a s t r o - ph . H E ] S e p (Kunz et al. 2014).The structure of this Letter is as follows. In §2, wesummarize the simulation setup and describe our treat-ment of viscosity. In §3, we present our main resultsregarding the morphology of the bubbles (§3.1) and theimpact on the ICM (§3.2). We discuss the implicationsof the results in §4 and conclude our findings in §5. METHODOLOGY
We perform 3D MHD simulations of one pair of jet-inflated AGN bubbles in a Perseus-like cluster usingFLASH (Fryxell et al. 2000). The simulation setup forthe initial ICM and magnetic field is identical to thatin Yang & Reynolds (2016a). The initial magnetic fieldis tangled with a coherence length of 25 kpc and theplasma beta ( β = p th /p B ) is ∼ . We choose a co-herence length that is greater than the typical size ofAGN bubbles because this is the optimal condition formagnetic draping to occur and help prevent bubble dis-ruption (Ruszkowski et al. 2008). The injection of AGNenergy in the simulations is purely kinetic, identical tothe KIN case in Yang et al. (2019). The AGN injectionhas a total jet power of × erg s − for a durationof 10 Myr, released along the ± z directions of the sim-ulation domain. Radiative cooling is omitted becausethe central cooling time of Perseus is longer than thesimulation duration (100 Myr). Four different assump-tions about the ICM viscosity are explored: (A) inviscid,(B) unsuppressed isotropic viscosity, (C) unsuppressedanisotropy viscosity, and (D) anisotropic viscosity lim-ited by the microscopic plasma instabilities.Viscosity in our simulations is included following themethod of ZuHone et al. (2015). Specifically, our simu-lations solve the following Braginskii-MHD equations: ∂ρ∂t + ∇ · ( ρ v ) = 0 , (1) ∂ ( ρ v ) ∂t + ∇ · (cid:16) ρ vv − BB π + p tot I (cid:17) = ρ g − ∇ · Π , (2) ∂E∂t + ∇· (cid:104) v ( E + p tot ) − B ( v · B )4 π (cid:105) = ρ g · v −∇· ( Π · v ) , (3) ∂B∂t + ∇ · ( vB − Bv ) = 0 , (4)where p tot = p + B / (8 π ) is the total pressure, and allother variables follow their usual definitions. The vis-cosity tensor for the isotropic case is defined as (Spitzer1962) Π iso = − µ ∇ v . (5)In the ICM, in which the gyro-radii of particles are muchsmaller than the Coulomb mean free path, the viscosityshould be anisotropic, and the viscosity stress tensor canbe expressed as (Braginskii 1965) Π aniso = − µ (cid:16) bb − I (cid:17)(cid:16) bb − I (cid:17) : ∇ v , (6)where µ = 2 . × − T / / ln Λ g cm − s − is thedynamic viscosity coefficient ( ln Λ = 30 ), and b is themagnetic field unit vector. For all viscous simulations, a ceiling is applied for the kinematic viscosity coefficient( ν = µ/ρ ) of cm s − . This is to prevent µ frombecoming unusually large within the bubbles due to hightemperatures resulted from purely kinetic jets. In Braginskii-MHD, the viscosity originates from thepressure anisotropy that arises due to conservation ofthe first and second adiabatic invariants of particles ontimescales that are much greater than the inverse of theion gyrofrequency (Chew et al. 1956). Under the condi-tion that the pressure anisotropy is balanced by its re-laxation via ion-ion collisions (Schekochihin et al. 2005),one can show that p ⊥ − p (cid:107) = 0 . p i ν ii ddt ln B ρ = 3 µ (cid:18) bb − I (cid:19) : ∇ v , (7)where p ⊥ and p (cid:107) are the pressure perpendicular andparallel to the magnetic field line, respectively, p i isthe ion thermal pressure, and ν ii is the ion-ion colli-sional frequency. The total thermal pressure satisfies p = (2 / p ⊥ +(1 / p (cid:107) . Given Eq. 7, the resulting viscousstress tensor could be written in a form identical to Eq. 6(see Section 3.1 of ZuHone & Roediger (2016) for a briefderivation). When the pressure anisotropy violates theinequalities − β (cid:46) ∆ p ≡ p ⊥ − p (cid:107) p (cid:46) β , (8)fast-growing firehose (which occurs when ∆ p < − /β )and mirror (when ∆ p > /β ) instabilities are triggeredand the pressure anisotropies should be kept within themarginal-stability thresholds (Schekochihin et al. 2005;Kunz et al. 2014). To account for this effect, in caseD we apply bounds to the pressure anisotropies (thuslimiting viscosity) according to Eq. 8.Note that the above means that the Braginskii-MHDequations (as used in case C and the previous study ofDong & Stone (2009)) become ill-posed when Eq. 8 isviolated. Without the microscopic effects being takeninto account, the fastest-growing modes of the instabil-ities formally occur at infinitely small scales, which areessentially the grid scale where the microinstabilities maybe unresolved. Later we will see that, indeed, while caseC is able to generate some modes of the firehose fluctu-ations, the mirror instability is not captured and hencethe positive pressure anisotropy could go beyond the sta-bility criterion, substantially overestimating the level ofviscosity. Although this case is rather unphysical, we in-clude it in this work in order to aid the interpretationof our results and to make a direct comparison with theprevious work of Dong & Stone (2009). RESULTS
Coherence of AGN bubbles
Figure 1 shows slices of gas density and temperatureat t = 60 Myr for cases A-D. In all four simulations,the energy injection from the AGN inflates bubbles thatare characterized by low densities and high temperatures.One immediately notices the different bubble morphol-ogy among the four simulations with different treatmentsof viscosity. For the inviscid simulation (A), the bub- Previous simulations of Reynolds et al. (2005) and Guo (2015)also used a constant µ to mitigate this effect. Figure 1.
Slices of gas density (top) and temperature (bottom) at t = 60 Myr for cases A (inviscid), B (unsuppressed isotropic viscosity),C (unsuppressed anisotropic viscosity), and D (anisotropic viscosity bounded by microinstabilities). ble shapes are irregular and the surfaces are rippled dueto Rayleigh-Taylor (RT) and Kelvin-Helmholtz (KH) in-stabilities as the low-density bubbles move through thedense ICM core with a velocity shear. As a result, thebubbles are gradually disrupted and mixed with the am-bient ICM. The timescales for the growth of RT and KHinstabilities at the bubble surface evaluated at t = 12 Myr are ∼ and 6.7 Myr, respectively (for density con-trast η ∼ . and shear velocity ∆ v ∼ km s − ).Note that in order for magnetic tension to suppress theKH instabilities, ∆ v has to be smaller than the rmsAlfvén speed in the two media (Chandrasekhar 1981).Given the large shear velocity at early times, magnetictension is unable to preserve the smooth surface of bub-bles self-consistently inflated by AGN jets.For the simulations with unsuppressed viscosity, eitherisotropic (B) or anistropic (C), the morphology of thebubbles is distinct from the inviscid case. Due to thesuppression of fluid instabilities by viscosity, the bubblesurface is much more smooth, and mixing is greatly in-hibited (evident from the bubble-ICM density and tem-perature contrasts). The importance of viscosity in casesB and C can be seen from Figure 2, which shows thatthe Reynolds numbers ( Re ≡ U L/ν ) are (cid:46) for thebubble interior. As a result, the bubbles look muchmore coherent in the mock X-ray image (bottom rowof Figure 2) in cases B and C, in contrast to the morepatchy bubbles with rippled surfaces in case A. Our resultfor the isotropic viscosity case confirms previous studies(Reynolds et al. 2005; Guo 2015). In contrast to Dong& Stone (2009), who showed that the coherence of bub-bles depends on magnetic field topology, we show thatunsuppressed anisotropic viscosity in tangled magneticfield can suppress the fluid instabilities and prevent thebubbles from disruption. Although anisotropic viscosityonly inhibits the instabilities along field lines, the ran-domness of the tangled field helps to stabilize the bubbles in multiple directions. Therefore, its effect is very sim-ilar to the isotropic case, though the effective isotropicviscosity is somewhat suppressed with respect to the fullSpitzer value due to the field geometry (analogous to thefactor of ∼ / − / suppression of thermal conductivity;Narayan & Medvedev 2001). This is also consistent withthe results of ZuHone et al. (2015), who found that theeffect on suppressing the KH instabilities for cold frontsof an isotropic viscosity ∼ − in the ambient medium. The pressureanisotropies are most significant at the shocks and withinthe bubbles, due to enhanced temperatures at these lo-cations. Without the microinstabilities, the pressureanisotropy in the bubble interior could reach ∼ . owingto significant compressive motions by the jets. Note thatthe pressure anisotropies are predominantly positive be-cause Braginskii-MHD simulations without the microin-stability limiter are able to capture the firehose instabil-ity (which regulates the negative pressure anisotropies)but not the mirror instability (see discussions in, e.g.,Kunz et al. 2012).On the other hand, the bubble interior is also where themagnetic field pressure is lower due to the adiabatic ex-pansion of the bubbles. Therefore, when the bounds forpressure anisotropies are applied, the enhanced plasmabeta within the bubbles ( β ∼ ) dramatically limits Figure 2.
Maps of the Reynolds number (top) and projected X-ray emissivity (bottom; fractional variation from a radially averagedprojected emissivity profile) at t = 60 Myr for cases A-D.
Figure 3.
Columns from left to right show slices of magnetic pressure overplotted with magnetic field vectors, plasma beta, pressureanisotropy ( ∆ p ), and departure from marginal stability ( f p ≡ β ∆ p ) for cases C (top) and D (bottom). the permitted range of pressure anisotropies and thusthe level of viscosity, to the degree that the bubblesare deformed by fluid instabilities just as in the invis-cid simulation. In contrast to the unlimited pressureanisotropies in case C that could go ∼ f p ≡ β ∆ p ∼ . In other words, the mi-croinstabilities effectively provide a factor of ∼ − suppression of the viscosity, thereby paralyzing its abilityto suppress the fluid instabilities. Impacts on the ICM
Figure 4 shows that evolution of radial profiles of theenclosed mass and the change of gas entropy ( K ≡ T /n / ), which traces ICM uplifting and locations ofheating by the bubbles, respectively. One can see that,except for the initial transient right after the jet injec-tion, the results could be divided into two groups: sim-ulations in which the bubbles are deformed (A and D),and simulations in which the bubbles maintain their in-tegrity (B and C). For the former group, the trailingpart of the bubbles tends to push the ICM outward atall radii within ∼ kpc, whereas the more coherentbubbles in the latter group are more capable of upliftingthe medium immediately surrounding the bubbles. Forthe former group, the heating primarily occurs in thewakes of the bubbles. This is where significant turbu-lent mixing takes place and the heating to the ICM is Figure 4.
Evolution of the enclosed mass profiles (top; normalized by radius squared to show the variation more clearly) and entropyprofiles relative to the initial state (bottom) for cases A-D.
Figure 5.
Time evolution of the maximum LOS velocity disper-sion smoothed with
Hitomi resolution for cases A-D. done by direct mixing with the ultra-hot bubbles (Yang& Reynolds 2016b). By contrast, for the latter group,the ICM is heated the most in regions surrounding thebubbles owing to both direct mixing and viscous heat-ing. In addition, the heat is deposited further away fromthe cluster center as the bubbles gradually move outwardand are disrupted on longer timescales.To compare the influence on the ICM kinematics bydifferent treatments of viscosity, we compute the line-of- sight (LOS) velocity dispersion ( σ LOS ≡ ( (cid:104) v l (cid:105)−(cid:104) v l (cid:105) ) / ,where brackets represent emission-weighted averages and v l is the velocity component along the LOS, which is as-sumed to be the x -axis here) smoothed with Hitomi res-olution. Figure 5 shows the maximum value across thegenerated σ LOS map at each epoch for all cases. In gen-eral, we do not find significant differences among differentsimulations regarding the overall evolution of σ LOS : allpeaks at ∼ − km s − within the first ∼ − Myr, and decreases to ∼ − km s − after 40 Myr.The LOS velocity dispersion is relatively insensitive toviscosity because the dispersion is dominated by fluidmotions on larger scales than the viscous scale (ZuHoneet al. 2018). Except for the first 20 Myr, these values areall smaller than the measured value by Hitomi of ∼ km s − in the NW region or ∼ km s − for other re-gions (Hitomi Collaboration 2018). Note, however, thatour simulations only considered a single AGN outburst.More realistic simulations of self-regulated AGN feed-back will be required to determine whether any of thesecases would generate too small velocity dispersions thatviolate observational constraints. DISCUSSION
Our results suggest that it remains challenging toproduce smooth and coherent bubbles by momentum-driven AGN jets. While magnetic tension could sup-press fluid instabilities for initially static bubbles (Robin-son et al. 2004; Ruszkowski et al. 2008), it is more dif-ficult to stabilize jet-inflated bubbles. While full Bra-ginskii anisotropic viscosity in a tangled magnetic fieldcould mimic isotropic viscosity and prevent bubble dis-ruption, when the pressure anisotropies are bounded bymicroinstabilities, the level of viscosity is dramaticallyreduced and the bubbles are deformed just as in theinviscid simulation. Therefore, our work suggests thatmicroinstability-limited Braginskii viscosity is unlikely tobe the primary mechanism for suppressing fluid instabil-ities for AGN bubbles.In order to explain the coherent structure and smoothsurface of observed bubbles, other mechanisms may berequired. For instance, Scannapieco & Brüggen (2008)showed that by including a subgrid model of turbulence,bubbles with smoother surfaces can be produced due tothe cancellation of small-scale modes of the fluid insta-bilities. Note, though, that their model of subgrid turbu-lence does not account for the KH instability, which is thedominant instability for bubbles inflated by momentum-driven jets. Moreover, the subgrid turbulence model em-ployed is based on ideal hydrodynamics, and whether ornot ideal hydrodynamic/MHD models are good approx-imations for the ICM remains an open question. Indeed,recent studies have shown that there exist fundamentaldifferences between MHD and Braginskii-MHD turbu-lence (Squire et al. 2019). Clearly, further studies are de-manded to understand the rich microphysics of the ICMplasma and how it impacts AGN feeding and feedbackon large scales.Though not the focus of our current study, here webriefly comment on the level of ICM viscosity in the re-gions excluding the bubbles. First of all, we find thatthe pressure anisotropies driven by shocks and soundwaves produced by the AGN outburst are around themarginal-stability threshold (rightmost column of Fig-ure 3), suggesting that the microinstabilities should haveminimal effects in these regions and the level of viscos-ity could be close to the full Braginskii value. Interest-ingly, this is consistent with constraints on the parallelviscosity obtained by observations of cold fronts (ZuHoneet al. 2015) and, assuming suppression due to fully tan-gled magnetic field, moderate suppression factors for theeffective isotropic viscosity ( f (cid:46) − ) inferred fromobservations of sloshing cold fronts and ram-pressurestripping tails of cluster galaxies (e.g., Su et al. 2017;Wang & Markevitch 2018). Such a non-negligible levelof viscosity would imply that viscous dissipation of soundwaves could still be an important source of AGN heat-ing (Fabian et al. 2003; Zweibel et al. 2018; Bambic &Reynolds 2019).Another implication of our work is that the viscosityof the ICM is likely highly spatially variable. While thelevel of viscosity is expected to be enhanced in regions ofhigh temperatures, it can be significantly limited by themicroinstabilities in regions with strong magnetic-fieldcompression/rarefactions. This stresses the importanceof Braginskii-MHD simulations with the microinstabilitylimiter in studies of ICM transport processes, includingcold-front and ram-pressure stripping simulations.There are a few limitations of our current work. First,our results only apply to kinetic-energy-dominated AGNjets. In reality, the composition of the jets and bubblesis largely unknown (Dunn & Fabian 2004). It remains tobe seen whether viscosity is required to preserve bubblesinflated by magnetically dominated jets (e.g., Li et al. 2006; O’Neill & Jones 2010), CR dominated jets (e.g.,Guo & Mathews 2011; Ruszkowski et al. 2017; Yang et al.2019), or internally subsonic jets in general (Guo 2016).In particular, magnetically dominated jets or other mech-anisms such as turbulent amplification (e.g., Yang et al.2013) could act to preserve/replenish the magnetic en-ergy within the bubbles, which may yield a lower plasmabeta within the bubbles and potentially alleviate the con-straint on viscosity. In addition, our simulations have ne-glected pre-existing turbulence in the ICM, which couldbreak one bubble into multiple segments and complicateobservational identification of radio bubbles (Heinz et al.2006). This effect should be taken into account whena more detailed comparison between simulated bubblesand observed cavities is performed in the future. CONCLUSIONS
We performed the first 3D Braginskii-MHD simula-tions of momentum-driven AGN jets to investigate theeffects of anisotropic viscosity on the evolution of bubblesthat are self-consistently inflated by the jets and evolve inrealistic, tangled magnetic field. We varied four differentcases of viscosity (inviscid, unsuppressed isotropic viscos-ity, unbounded anisotropic viscosity, anisotropic viscositybounded by microinstabilities) to study the morphologyof the bubbles as well as its resulting impacts on theICM. Our conclusions are as follows.1. When jet inflation of the bubbles is self-consistentlymodeled, even magnetic field with coherence lengthsgreater than the bubble size cannot prevent the defor-mation of the bubbles.2. Unsuppressed anisotropic viscosity along tangledmagnetic field lines can have similar effects as isotropicviscosity and is capable of preventing the bubbles fromdisruption. However, the level of viscosity in the vicinityof the bubbles in this case is overestimated by a factor of ∼ − compared to the case with bounded pressureanisotropies.3. Adding bounding microinstabilities to the pressureanisotropy of the system drastically changes the outcomeof the bubble evolution. The viscosity within the bub-bles is so significantly suppressed by the microinstabili-ties that it can no longer prevent the bubbles from defor-mation, resembling the inviscid case. Note that the con-straints on viscosity depend on the plasma β within thebubbles and thus could potentially be alleviated by mech-anisms that could enhance the bubble magnetic field.4. The LOS velocity dispersions computed from thecurrent simulations saturate at ∼ − km s − forall cases, which is smaller than the Hitomi measurementfor the NW region. Future simulations of self-regulatedfeedback are required to determine whether any of thesemodels can be ruled out by observational data from
Hit-omi, XRISM, Athena and
Lynx .5. The ability to uplift the ICM and the locations ofheating by the bubbles critically depend on whether ornot the bubbles are preserved. In other words, obtainingan accurate prescription for the bubble-ICM interactionremains a key question for determining the dynamicalimpact and heating of the ICM by AGN jets.Our simulations suggest that it remains a challenge toproduce smooth and coherent bubbles as the NW ghostcavity in Perseus by momentum-driven AGN jets; mech-anisms other than Braginskii viscosity appear to be re-quired to suppress the fluid instabilities at bubble sur-faces. Detailed comparisons between Braginskii-MHDsimulations and high spatial and spectral resolution X-ray observations of AGN bubbles, ICM turbulence, coldfronts, and ram-pressure stripping tails will provide cru-cial constraints on the transport coefficients of the ICM.Our results also highlight the dramatic influence of the“microphysics" on the macroscopic properties of AGNbubble evolution. Accurate modeling of the ICM plasmais thus fundamental for constructing a robust model forAGN feeding and feedback.
ACKNOWLEDGEMENTS
H.Y.K.Y. acknowledges support from NASA ATP(NNX17AK70G) and NSF (AST 1713722). J.A.Z. ac-knowledges support from NASA contract NAS8-03060with the
Chandra
X-ray center. The simulations wereperformed on
Pleiades at NASA and