Sloshing of the Magnetized Cool Gas in the Cores of Galaxy Clusters
aa r X i v : . [ a s t r o - ph . C O ] S e p Draft version September 22, 2018
Preprint typeset using L A TEX style emulateapj v. 03/07/07
SLOSHING OF THE MAGNETIZED COOL GAS IN THE CORES OF GALAXY CLUSTERS
J. A. ZuHone , M. Markevitch , D. Lee Draft version September 22, 2018
ABSTRACTX-ray observations of many clusters of galaxies reveal the presence of edges in surface brightnessand temperature, known as “cold fronts”. In relaxed clusters with cool cores, these edges have beeninterpreted as evidence for the “sloshing” of the core gas in the cluster’s gravitational potential. Thesmoothness of these edges has been interpreted as evidence for the stabilizing effect of magnetic fields“draped” around the front surfaces. To check this hypothesis, we perform high-resolution magne-tohydrodynamics simulations of magnetized gas sloshing in galaxy clusters initiated by encounterswith subclusters. We go beyond previous works on the simulation of cold fronts in a magnetizedintracluster medium by simulating their formation in realistic, idealized mergers with high resolution(∆ x ∼ Subject headings: galaxies: clusters: general — X-rays: galaxies: clusters — methods: hydrodynamicsimulations INTRODUCTION
Observations with the most recent generation of X-ray telescopes have shown that clusters looking “re-laxed” upon first glance often have cool cores that arein a disturbed state. Many “cool-core” systems exhibitedges in X-ray surface brightness approximately concen-tric with respect to brightness peak of the cluster (e.g.,Mazzotta et al. 2001; Markevitch et al. 2001, 2003). X-ray spectra of these regions have revealed that in mostcases the brighter (and therefore denser) side of the edgeis the colder side, and hence these jumps in gas den-sity have been dubbed “cold fronts” (for a detailed re-view see (Markevitch & Vikhlinin 2007)). The presenceof one or more cold fronts is an indication of the mo-tion of gas in a cluster. Famous examples of cold frontsformed by ongoing major mergers are those in Abell3667 (Vikhlinin et al. 2001) and the “Bullet Cluster”(1E0657-56) (Markevitch et al. 2002). However, coldfronts also occur in clusters which appear relaxed on thelargest scales and show no obvious indications of ma-jor merging (e.g. Abell 1795 and Abell 2029). Thesefronts can arise due to “sloshing” motion of the coregas in the dark-matter dominated gravitational poten-tial. This process was studied in detailed hydrodynamicsimulations in Ascasibar & Markevitch (2006) (hereafterAM06), Tittley & Henriksen (2005) (though with a dif- Astrophysics Science Division, Laboratory for High Energy As-trophysics, Code 662, NASA/Goddard Space Flight Center, Green-belt, MD 20771 Smithsonian Astrophysical Observatory, Harvard-SmithsonianCenter for Astrophysics, Cambridge, MA 02138 Department of Astronomy, ASC Flash Center, University ofChicago, 5747 S. Ellis Avenue, Chicago, IL 60637 ferent interpretation), ZuHone et al. (2010) (hereafterZMJ10), and Roediger et al. (2011). In these studies,the sloshing arises from encounters with small infallinggroups or subclusters that gravitationally perturb thecluster core. Since the gas core is subject to ram pressurebut the dark matter core is not, such perturbations canresult in a separation between the gas and dark matterpeaks, and consequently the core gas will begin to “slosh”back and forth in the gravitational potential well.In ZMJ10, who used high-resolution grid-based simu-lations, it was demonstrated that if the ICM is modeledas an unmagnetized, inviscid fluid, the initially smoothand sharp cold fronts that result from sloshing motionsare quickly disrupted by Kelvin-Helmholtz instabilitiescaused by the large shearing velocities present across thecold fronts. Such instabilities make the fronts appearragged and torn, in contrast to the fronts that are seenin X-ray observations, which appear smooth and sharp.Additionally, mixing in these simulations was very effi-cient, resulting in the heating of the initially cool gasin the core as it was mixed with hotter gas brought intocontact by sloshing. Conversely, ZMJ10 also showed thatif the ICM is viscous, the instabilities are damped outand the resulting cold fronts retain their smooth, sharpshape. The smoothness of observed cold fronts in clus-ters of galaxies indicates that the ICM might possess asignificant viscosity. An additional effect of viscosity inthe ZMJ10 simulations was to suppress mixing of gasesof different entropies, preventing cluster cool cores fromheating due to such mixing.An alternative mechanism for the stability of coldfronts, which has been discussed previously in the litera-ture, is the effect of magnetic fields. Strong observationalevidence points to the existence of magnetic fields per-meating the cluster volume (see Carilli & Taylor 2002;Ferrari et al. 2008, for recent reviews). One such line ofevidence is synchrotron radio emission from sources suchas radio halos (Feretti et al. 2001; Govoni et al. 2001),radio mini-halos in the cluster cool cores (Burns et al.1992; Bacchi et al. 2003; Venturi et al. 2007; Gitti et al.2007; Govoni et al. 2009; Giacintucci et al. 2011), andradio relics. It is possible to estimate the magnetic fieldof the ICM under the assumption of equipartition be-tween the relativistic electron population and the mag-netic field (Pacholczyk 1970). The magnetic field canalso be estimated by comparing the synchrotron radioemission with inverse Compton hard X-ray emission fromthe same relativistic electrons, without the need to as-sume equipartition between the particles and the field.However, nonthermal X-ray emission from clusters hasso far eluded confident detection–early reports of excessover the cluster thermal emission at high energies (Coma,Fusco-Femiano et al. 2004; A2163, Rephaeli et al. 2006)either allow an alternative thermal explanation or con-tradict the more recent upper limits from the Suzakuand SWIFT telescopes (e.g., Nakazawa et al. 2009; Sug-awara et al. 2009; Ajello et al. 2009; Wik et al. 2009).Nondetection of inverse Compton emission correspondsto lower limits on the average magnetic field strengthsof 0 . − . µ G in radio halo regions (e.g., Ajello et al.2009; Wik et al. 2009) and a few µ G in relic regions(e.g., Finoguenov et al. 2010), broadly consistent withthe equipartition radio estimates and Faraday rotationdata.A second line of evidence for magnetic fields in clus-ters is that of Faraday rotation of polarized emission ofradio sources. Rotation measure (RM) studies indicatethat magnetic field strengths in clusters are on the orderof a few µ G, with strengths up to tens of µ G in clustercool cores (Perley & Taylor 1991; Taylor & Perley 1993;Feretti et al. 1995, 1999; Taylor et al. 2002, 2006, 2007;Bonafede et al. 2010). Additionally, in some clusters itis possible to derive RM maps, which are typically quitepatchy, indicating that the coherence length of the clus-ter magnetic field is on the order of 10 kpc or less. High-resolution RM maps have been used to infer the clus-ter magnetic field power spectrum (Vogt & Enßlin 2003;Murgia et al. 2004; Vogt & Enßlin 2005; Govoni et al.2006; Guidetti et al. 2008; Govoni et al. 2010). Thesestudies indicate that the magnetic field power spectrumis similar to a Kolmogorov type ( P B ( k ) ∝ k − / ), de-pending on the assumed value for the coherence lengthof the field fluctuations.Magnetic fields oriented parallel to a shearing surfacewill suppress the growth of the Kelvin-Helmholtz in-stability (Landau & Lifshitz 1960; Chandrasekhar 1961).Whether or not the stability of cold fronts can be pro-vided by magnetic fields depends on the strength of thefield and the orientation of the field with respect to thefront surface. What is required is a field oriented par-allel to the front surface which has a magnetic pres-sure comparable to the kinetic energy per unit volumeof the shearing flow across the front. For a cold frontin the galaxy cluster A3667, Vikhlinin et al. (2001) andVikhlinin & Markevitch (2002) (hereafter V01 and V02)determined that the magnetic field strength required to stabilize the front is B ∼ µ G, roughly an order of mag-nitude higher than the field strengths usually inferredfrom RM estimates and synchrotron diffuse radio emis-sion outside of the cooling core regions. Additionally, theindications from RM maps that the magnetic field is tan-gled with a small coherence length seems to argue againstthe existence of a large-scale field that could drape a coldfront.However, the region surrounding a cold front is nota typical place in a galaxy cluster. As argued inV01/V02, a cold front moving through the intraclus-ter medium will cause the flow of the surrounding ICMto move around it, creating a shear flow. Lyutikov(2006) pointed out that such flows around cold frontsand radio bubbles in galaxy clusters lead to “mag-netic draping.” Provided the motion of the front issuper-Alfvenic, a weak, tangled magnetic field will bestretched by this shear flow to produce a layer paral-lel to the the front surface. The magnetic field energyin this layer will be increased due to shear amplifica-tion, possibly strong enough to stabilize the front againstKelvin-Helmholtz instabilities (Keshet et al. 2010). Anumber of previous simulation works (e.g Asai et al.2004, 2007; Dursi 2007; Dursi & Pfrommer 2008) havedemonstrated the stabilizing effect of magnetic fields forcold fronts and AGN-blown bubbles in simplified situa-tions, where cold “blobs” and hot bubbles propagatedin simple, typically stratified atmospheres and differ-ent field geometries (e.g., uniform, tangled) were con-sidered. Takizawa (2008) simulated more realistic merg-ers with N -body/magnetohydrodynamics simulations,showing that magnetic fields wrapped around mergercold fronts (such as in the Bullet Cluster) and stabilizedthem against instabilities. Here, we extend these stud-ies by simulating the formation of sloshing cold frontsproduced by an encounter of a relaxed galaxy clusterwith a small, infalling subcluster. Our aim is to deter-mine for realistic magnetic field strengths and configu-rations whether or not the magnetic fields are amplifiedand drape the cold fronts to a sufficient extent to sup-press instabilities that would otherwise grow and disruptthe smoothness of the fronts.The interplay between gas sloshing motions and mag-netic fields in galaxy clusters may have other interestingimplications. Observations of clusters with radio mini-halos in their cool cores revealed a spatial associationbetween sloshing cold fronts and the observed radio emis-sion (Mazzotta & Giacintucci 2008), suggesting that theradio emission originates from the effects of the sloshingmotions. This may come from turbulence generated bythe sloshing motions, the amplification of magnetic fieldstrengths due to the associated shear flows (Keshet et al.2010), or a combination of both. Another interestingobservable effect is the large, amplified magnetic fieldstructures produced by sloshing may produce signaturesin RM maps associated with the sloshing structures. Full3D MHD simulations of sloshing such as ours may deter-mine if such effects may be associated with cold fronts inrelaxed clusters.This is a first paper in a series describing simulationsof magnetized gas sloshing in galaxy cluster cores. Thepurposes of this work are to a) give technical details ofthe simulation setup, b) present results on the effect ofsloshing of the gas and magnetic fields of the cluster,and c) serve as the groundwork for future studies whichwill explore additional physics and observable effects ofsloshing. This paper is organized as follows: in Section2 we describe the characteristics of the simulations andthe code. In Section 3 we describe the characteristicsof gas sloshing in our simulations in the absence andpresence of magnetic fields, in particular with regard tothe effect of sloshing on the fields and the effect of thefields on the structure of the cold fronts. In Section 4we discuss the implications of these results. Finally, inSection 5 we summarize our results and discuss futuredevelopments of this work. Throughout this paper weassume a flat ΛCDM cosmology with h = 0 .
7, Ω m = 0 . b = 0 . h − . SIMULATIONS
Method
In our simulations, we solve the ideal MHD equations.Written in conservation form in Gaussian units, theyare: ∂ρ∂t + ∇ · ( ρ v ) = 0 (1) ∂ ( ρ v ) ∂t + ∇ · ( ρ vv − BB ) + ∇ p = ρ g (2) ∂E∂t + ∇ · [ v ( E + p ) − B ( v · B )] = ρ g · v (3) ∂ B ∂t + ∇ · ( vB − Bv ) = 0 (4)where p = p th + B π (5) E = ρv ǫ + B π (6)where p th is the gas pressure, and ǫ is the gas internal en-ergy per unit volume. For all our simulations, we assumean ideal gas equation of state with γ = 5 / N -body astrophysical simula-tion code developed at the Center for AstrophysicalThermonuclear Flashes at the University of Chicago(Fryxell et al. 2000; Dubey et al. 2009). FLASH usesadaptive mesh refinement (AMR), a technique thatplaces higher resolution elements of the grid only wherethey are needed. We are interested in capturing sharpICM features like shocks and cold fronts accurately, aswell as resolving the inner cores of the cluster dark matterhalos. It is particularly important to be able to resolvethe grid adequately in these regions. AMR allows us todo so without needing to have the whole grid at the sameresolution.FLASH 3 solves the equations of magnetohydrody-namics using a directionally unsplit staggered meshalgorithm (USM; Lee & Deane 2009). The USM al-gorithm used in FLASH 3 is based on a finite-volume, high-order Godunov scheme combined witha constrained transport method (CT), which guar-antees that the evolved magnetic field satisfies thedivergence-free condition (Evans & Hawley 1988). Inour simulations, the order of the USM algorithm cor-responds to the Piecewise-Parabolic Method (PPM) ofColella & Woodward (1984), which is ideally suited for capturing shocks and contact discontinuties (such as thecold fronts that appear in our simulations). The USMsolver in FLASH comes equipped with two methods forinterpolating magnetic fields in a divergenceless mannerfrom coarse to fine blocks on the AMR grid. The firstis by simple injection of the magnetic field componentsfrom the coarser cells to the finer neighboring cells. Thesecond is a higher-order method described in Balsara(2001). We find that the choice of either prescriptiondoes not affect our conclusions.The gravitational potential on the grid is set up as thesum of two “rigid bodies” corresponding to the contribu-tions to the potential from both clusters. This approachto the modeling the potential is used for simplicity andspeed over solving the Poisson equation for the matterdistribution, and is an adequate approximation for ourpurposes. It is the same approach that we used in ZMJ10for the resolution test; it will be described in detail inSection 2.2. Initial Conditions
Our initial conditions in these idealized simulationshave been set up in a manner very similar to ZMJ10,with some differences which we elaborate on here.For the cluster dark matter profile we have chosen aHernquist (1990) profile: ρ DM ( r ) = M tot πa r/a )(1 + r/a ) (7)where M tot and a are the mass and scale length of the DMhalo. The Hernquist profile shares with the more com-monly employed Navarro, Frenk, & White (1997, NFW)profile a “cuspy” inner radial dependence of the darkmatter density, but results in simpler expressions for themass, potential, and particle distribution functions. Be-cause we are interested in the consequences of the inter-action for only the central regions of the main cluster,the difference in the density dependence for large radii isunimportant. The corresponding gravitational potentialhas a particularly simple form:Φ DM ( r ) = − GM tot r + a (8)which for r ≫ a behaves as a point mass potential and for r ≪ a is approximately constant. For our simulations,the same profile shape is used for both the main clusterand the subcluster.For the gas temperature, we use a phenomenologicalformula: T ( r ) = T r/a c + r/a c r/a c (9)where 0 < c < a c is the characteristic radius of that drop. This func-tional form can reproduce cluster temperature profilesof many observed relaxed galaxy clusters, which havea characteristic temperature drop in the center due tocooling. With this temperature profile, the correspond-ing gas density can be derived by imposing hydrostaticequilibrium: ρ gas ( r ) = ρ (cid:18) ra c (cid:19) (cid:18) r/a c c (cid:19) α (cid:16) ra (cid:17) β , (10)with exponents α ≡ − − n c − c − a/a c , β ≡ − n − a/a c c − a/a c . (11)We set n = 5 in order to have a constant baryon frac-tion at large radii, and we compute the value of ρ fromthe constraint M gas /M DM = Ω gas / Ω DM . This gas profileresembles those of most cool-core clusters since it contin-ues to increase with decreasing radius and does not havea flat core. From these profiles we can derive a radialdependence for the cluster entropy profile: S ≡ k B T n − / e ∝ (cid:18) ra c (cid:19) − / (cid:18) r/a c c (cid:19) (3 − α ) / (cid:16) ra (cid:17) − (2 β +3) / (12)which resembles a power-law S ( r ) ∝ r . − . over mostof the radial range, in line with observations of cool-coreclusters (e.g., Donahue et al. 2006). The initial radialprofiles for the main cluster are given in Figure 1.Our merging clusters consist of a large, “main” cluster,and a small infalling subcluster. They are characterizedby the mass ratio R ≡ M /M , where M = M R/ (1+ R )and M = M / (1 + R ) are the masses of the main clusterand the infalling satellite, respectively. The subclusterpotential center starts at a distance d from the maincluster center, and with an initial impact parameter b .For all of our simulations in this parameter study, wechoose the same subcluster mass ratio ( R = 5), distance d = 3 Mpc, and impact parameter b = 500 kpc. To scalethe initial profiles for the various mass ratios of the clus-ters, the combinations M i /a i , c i , and a c,i /a i are heldconstant. For the main cluster, we chose a = 600 kpc, c = 0.17, and a c, = 60 kpc, to resemble mass, gas den-sity, and temperature profiles typically observed in realgalaxy clusters. In particular, our main cluster closelyresembles A2029, a hot, massive, relaxed cluster with acool core that exhibits a cold front (e.g., Vikhlinin et al.2005, 2006; Ascasibar & Markevitch 2006). For all of thesimulations, we set up the main cluster within a cubicalcomputational domain of width L = 2 . x = 2 .
34 kpc(see Appendix B for the results of a resolution test).The initial cluster velocities are chosen so that the totalkinetic energy of the system is set to a fraction 0 ≤ K ≤ E ≈ ( K − GM M d = ( K − R (1 + R ) GM d (13)So the initial velocities in the reference frame of the cen-ter of mass are set to v = R √ K R r GM d ; v = √ K R r GM d (14)For the simulations presented in this work, we have set K = 1 / x and velocity v is integrated over atimestep ∆ t using the variable-timestep leapfrog methodoften used for integration of particles in N -body simula-tions (Hockney & Eastwood 1988): x i = x i + v i ∆ t (15) v / i = v i + 12 a i ∆ t (16) v n +1 / i = v n − / i + C n a ni + D n a n − i (17) x n +1 i = x ni + v n +1 / i ∆ t n (18)with the coefficients C n and D n given by C n = 12 ∆ t n + 13 ∆ t n − + 16 (cid:20) (∆ t n ) ∆ t n − (cid:21) (19) D n = 16 (cid:20) ∆ t n − − (∆ t n ) ∆ t n − (cid:21) (20)where n is the time index, i is the spatial index, and a is the particle’s acceleration. By using time-centeredvelocities and stored accelerations, this method achievessecond-order time accuracy.The gravitational potential at all points on the gridis assumed to be the sum of the respective potentials ofthe main cluster and the subcluster. The gravitationalacceleration is then computed by finite differencing thepotential. Since the frame we have chosen (with the maincluster fixed at the center) is not an inertial frame, wemust also compute the inertial acceleration on the maincluster from the subcluster and add it to the accelerationfrom gravity.In a real cluster merger, the subcluster will be tidallystripped of its dark matter, growing smaller in mass, and(if it is bound to the main cluster) each subsequent pas-sage will induce a correspondingly weaker disturbanceon the main cluster core, until the subcluster is fullyabsorbed into the main cluster. Additionally, the sub-cluster’s orbit would be altered due to the effects of dy-namical friction from the surrounding dark matter of themain cluster. In our simplified model, the subcluster isnot stripped of its mass, and there is no dynamical fric-tion, so it travels on a closed orbit which would eventu-ally take it on the same trajectory past the main cluster.This would result in a second disturbance that would beequal in magnitude to the original passage and be highlydisruptive. In order to study the effects of the result-ing sloshing in isolation without the interference of sub-sequent crossings of the subcluster, the aforementionedtrajectory is followed until the center of the subclusterreaches the opposite side of the box, at which point itis assumed to follow a constant-velocity trajectory forthe remainder of the simulation. As was seen in ZMJ10,that used a more realistic N -body representation of theclusters’ DM components, the majority of the effects onthe main cluster from the subcluster are due to the firstcore passage, so the essential features of the encounterare still captured.This gravitational potential setup is used for compu-tational simplicity, and is adequate for our qualitativestudy of hydrodynamic effects. We find that this pro-cedure for computing the trajectory of the subclusters’ TABLE 1Simulation Parameter Space
Simulation Initial β Initial Field Configuration Initial λ (kpc)NoFields N/A N/A N/ABeta100 100 Tangled 43Beta400 400 Tangled 43Beta1600 1600 Tangled 43Beta6400 6400 Tangled 43Tight 400 Tangled 15Loose 400 Tangled 120Tangential 400 Tangential N/A orbit and the total acceleration on the gas reproduceswell the general characteristics of the sloshing featuresseen in AM06 and ZMJ10. A forthcoming paper de-tailing quantitatively the similarities and differences be-tween the self-gravitating and the rigid potential sloshingsetups supports our use of the rigid-potential approxima-tion (Roediger & ZuHone 2011).Finally, it remains to set up the magnetic field ofthe cluster. For a realistic cluster magnetic field, itis important to satisfy a few basic conditions. Thefirst is the magnitude of the field itself. Typical mea-surements from Faraday rotation and synchrotron radia-tion measurements suggest field strengths of of 1-10 µ G.This implies that the plasma β for the magnetic field( β = p/p B , where p B = B / π ) is high, with typicalvalues in the range 100-1000 (e.g., a field of 3 µ G at r =150 kpc in Abell 2029 corresponds roughly to β ≈ ∇ · B = 0.In order to have these two conditions simultane-ously satisfied, we use the following procedure, as inRuszkowski et al. (2007) and Ruszkowski & Oh (2010).A random magnetic field ˜ B ( k ) is set up in k -space on auniform grid using independent normal random deviatesfor the real and imaginary components of the field. Thus,the components of the complex magnetic field in k -spaceare set up such that˜ B x ( k ) = B [ N ( u ) + iN ( u )] (21)˜ B y ( k ) = B [ N ( u ) + iN ( u )] (22)˜ B z ( k ) = B [ N ( u ) + iN ( u )] (23)where N ( u ) is a function of the uniformly distributedrandom variable u that returns Gaussian-distributed ran-dom values, and the values B i are field amplitudes. Weadopt a dependence of the magnetic field amplitude B ( k )on the wavenumber | k | similar to (but not the same as)Ruszkowski et al. (2007) and Ruszkowski & Oh (2010): B ( k ) ∝ k − / exp[ − ( k/k ) ]exp[ − k /k ] (24)where k and k control the exponential cutoff termsin the magnetic energy spectra. The cutoff at highwavenumber k roughly corresponds to the coherencelength of the magnetic field k = 2 π/λ (e.g., the scaleof the observed patches in the RM maps, see ) and wevary this for a few of our simulations. The cutoff at lowwavenumber k = 2 π/λ roughly corresponds to λ ≈ r / ≈
500 kpc, which is held fixed for all of our simula-tions. This field spectrum corresponds to a Kolmogorovshape for the energy spectrum ( P B ( k ) ∝ k − / ) with cutoffs at large and small linear scales. This field is thenFourier transformed to yield B ( x ), which is rescaled tohave an average value of p πp/β to yield a field that hasa pressure that scales with the gas pressure, i.e. to havea spatially uniform β for the initial field. For our simula-tions, we try different values of β to determine the effectsof different initial field strengths, and different values of λ to determine the effects of different initial field con-figuration. There are also predictions from simulationswith magnetic fields and anisotropic heat conductionthat in the cluster cool cores, the magnetic field orienta-tion may be preferentially tangential due to the heat-fluxbouyancy instability (HBI, see, e.g., Parrish & Quataert2008; Bogdanovi´c et al. 2009; Parrish et al. 2009). Forthis reason, it would be interesting to examine the effectof an initial field configuration where the field lines werepreferentially tangential. For this purpose, we include asimulation that has a purely tangential initial field sim-ilar to that used in Bogdanovi´c et al. (2009), describedin spherical coordinates by B r = 0 (25) B θ = 2 B sin θ cos 2 φ (26) B φ = − B sin 2 φ sin 2 θ (27)Its magnitude was then scaled by the gas pressure.The entire set of simulations is summarized in Table 1.Finally, for all configurations, a field consistent with ∇ · B = 0 is generated by “cleaning” the field of divergenceterms in Fourier space via˜ B ( k ) → ( I − ˆkˆk ) ˜B ( k ) (28)where ˆ k is the unit vector in k -space, and I is the identityoperator. Upon transformation back to real space, themagnetic field satisfies ∇ · B = 0. This field is theninterpolated onto our AMR grid in such a way that thecondition ∇ · B = 0 is maintained.Finally, to test the robustness of our initial model forthe main cluster, we perform a test run with the maincluster kept unperturbed, with an average magnetic fieldstrength near β = 100-200. Figure 1 shows the profiles ofgas density, gas temperature, and gas entropy at the be-ginning of the simulation and at a later epoch ( t = 5 Gyrafter the beginning of the simulation), demonstrating thestability of these cluster quantities at all radii exceptingthe innermost couple of resolution elements (of width ∼ Fig. 1.—
Radial profiles of gas density, gas temperature, gaspressure, gas entropy, magnetic field strength, and plasma β at theepochs t = 0.0 Gyr and t = 5.0 Gyr for a single cluster evolved inisolation. magnetic field itself does evolve to a slightly differentconfiguration, an effect we will discuss in Appendix A. RESULTS
Sloshing of the Cool Core in an Unmagnetized TestCase
First, we will briefly describe the sloshing process dueto subcluster mergers as elucidated in Section 3 of AM06and Section 3 of ZMJ10. This general description is ap-plicable to all of our simulations, regardless of the detailsof the initial magnetic field or its presence or absence, butwe will be referring in this description to our “control”simulation
NoFields , in which there is no magnetic fieldin the cluster gas.It is assumed that the subcluster has lost its gas dueto ram pressure stripping from an earlier phase of themerger (although as the subcluster approaches the maincluster it begins to drag some of the cluster’s ICM in atrailing sonic wake). This is seen in the first panel of Fig-ure 2, which shows slices of temperature through the clus-ter center in the plane of the clusters’ mutual orbit. Thecore passage of the subcluster occurs at approximately t ∼ . t = 5 . Fig. 2.—
Slices through the gas temperature for the simulationwith no fields at various epochs. Each panel is 500 kpc on a side.Major tick marks indicate 100 kpc distances. The color scale showstemperature in keV. The black dot marks the position of the clusterpotential minimum. it. In addition to the gravitational disturbance, the waketrailing the subcluster transfers some of the angular mo-mentum from the subcluster to the core gas and also actsto help push the core gas out of the DM potential well.As the cool gas from the core climbs out of the po-tential minimum, it expands adiabatically. However, thedensest, lowest-entropy gas quickly begins to sink backtowards the potential minimum against the ram pressurefrom the surrounding ICM. Once again, as the cool gasfalls into the potential well it overshoots it, and the pro-cess repeats itself on a smaller linear scale. Each time, acontact discontinuity (“cold front”) is produced. Due tothe angular momentum transferred from the subclusterby the wake, these fronts have a spiral-shaped structure.Throughout this process, higher-entropy gas from largerradii is brought into contact with the lower entropy gasfrom the core, and as these gases mix, the entropy of thecore gas is increased (ZMJ10). The last panel of Fig-ure 2 shows that by the epoch t = 4 . NoFields are qualitatively very similiar to the inviscid results fromZMJ10 (the only difference is the way the gravitationalpotential is modeled, which in that case was computedfrom the self-gravity of the DM and gas components ofthe cluster). The fronts start off very smooth, but quicklydevelop large Kelvin-Helmholtz-driven billows, due tothe large shear flows that are present across the frontsurfaces. The action of the instabilities on the fronts isto make them appear jagged, and eventually even disap-pear (instead of the smooth fronts seen in observations).
The Effect of Sloshing on the Magnetic Field
We explore our parameter space by varying two condi-tions of the magnetic field: the initial average strength ofthe magnetic field and the initial spatial configuration ofthe magnetic field. We will discuss the evolution of themagnetic field in these two sets of simulations separately.All of the simulations begin with a roughly uniform
Fig. 3.—
Slices through the magnetic field strength with vectors indicating field direction in the x-y plane for the
Beta6400 simulation.Each panel is 500 kpc on a side. Major tick marks indicate 100 kpc distances. The color scale shows magnetic field strength in µ G. Fig. 4.—
Slices through the magnetic field strength with vectors indicating field direction in the x-y plane for the
Beta1600 simulation.Each panel is 500 kpc on a side. Major tick marks indicate 100 kpc distances. The color scale shows magnetic field strength in µ G. Fig. 5.—
Slices through the magnetic field strength with vectors indicating field direction in the x-y plane for the
Beta400 simulation.Each panel is 500 kpc on a side. Major tick marks indicate 100 kpc distances. The color scale shows magnetic field strength in µ G. Fig. 6.—
Slices through the magnetic field strength with vectors indicating field direction in the x-y plane for the
Beta100 simulation.Each panel is 500 kpc on a side. Major tick marks indicate 100 kpc distances. The color scale shows magnetic field strength in µ G. Fig. 7.—
The alignment of cold fronts with strongly magnetized layers. Top panels: Temperature (keV, left) and magnetic field strength( µ G, right) for the
Beta100 simulation at the epoch t = 3.15 Gyr. Bottom panels: Temperature (keV, left) and magnetic field strength( µ G, right) for the
Beta400 simulation at the epoch t = 3.75 Gyr. Vectors indicate magnetic field direction in the x-y plane. Each panelis 500 kpc on a side. β with small fluctuations. Since the measured fieldstrengths in clusters of galaxies is uncertain by an or-der of magnitude or so, it is important to characterizethe effect of varying the initial strength of the magneticfield on its subsequent evolution. The simulation set { Beta100,Beta400,Beta1600,Beta6400 } explores the ef-fects of varying the initial magnetic field strength for arange of field strengths that covers the currently obser-vationally plausible interval. The initial magnetic fieldconfiguration for each of these simulations is the same,corresponding to a Kolmogorov power spectrum with k = 2 π/ (43 kpc) and k = 2 π/ (500 kpc) (see Section2.2).The sloshing motions drag the field around, amplify-ing it along shear flows. Figures 3 through 6 illustratethe effect of sloshing on the magnetic field strength anddirection over the course of time. The first panel ineach figure shows the tangled magnetic field before the sloshing motions begin, with the radially averaged fieldstrength highest at the center and steadily decreasingoutward. Over the course of the next few Gyr as rep-resented in the following panels, the sloshing motionsamplify the field and increase the size of the stronglymagnetized region. These fields are also ordered on largescales, aligned largely along the cold front surfaces andother places in the domain where there are shear flows.In these layers the field strengths can be amplified upto tens of µ G. Within the envelope of the cold fronts, atlater epochs the fields are once again tangled, though thefield strengths have been increased by a factor of ∼ Beta100 simulation result inmagnetic field strengths that is over ∼ µ G out to radiiof r ∼
150 kpc (Figure 6), potentially in conflict with ob-0
Fig. 8.—
Slices through the plasma β for the simulations withvarying initial β at the epoch t = 2.5 Gyr. Each panel is 500 kpcon a side. Major tick marks indicate 100 kpc distances. Fig. 9.—
Slices through the plasma β for the simulations withvarying initial β at the epoch t = 3.15 Gyr. Each panel is 500 kpcon a side. Major tick marks indicate 100 kpc distances. servations. However, in each simulation we do not findany magnetic field strengths in excess of tens of µ G, im-plying that the fields will not increase without limit butwill saturate.Figure 7 shows side-by-side examples of the tempera-ture and the magnetic field strength for two simulationsat two different epochs (with magnetic field vectors over-laid), demonstrating the alignment of the strongly mag-netized layers with the cold fronts. These layers are sit-uated right underneath the cold front surfaces, and thefield lines along these layers are stretched along the di-rection of the fronts, in agreement with the expectationfrom Keshet et al. (2010).Figures 8 through 11 show slices of the plasma β through the center of the domain for the epochs t =2.5, 3.15, 3.75, and 4.5 Gyr after the beginning of the Fig. 10.—
Slices through the plasma β for the simulations withvarying initial β at the epoch t = 3.75 Gyr. Each panel is 500 kpcon a side. Major tick marks indicate 100 kpc distances. Fig. 11.—
Slices through the plasma β for the simulations withvarying initial β at the epoch t = 4.5 Gyr. Each panel is 500 kpcon a side. Major tick marks indicate 100 kpc distances. simulation, for the simulations with varying initial β . Ineach case, the shear flows amplify the magnetic field, de-creasing β . The degree to which β is decreased in theselayers is dependent on the initial β , however in each casethe degree of amplification of the field energy is similar,on an order of magnitude or higher.Our default setup for the magnetic field spatial config-uration, as detailed in Section 2.2, is a tangled magneticfield. This field may be characterized by the slope of theinitial power spectrum (which we take to be Kolmogro-rov), the cutoff at large k ( k ), and the cutoff at small k ( k ). It is particularly instructive to examine the ef-fect of varying the cutoff at large k (small scales), sincea magnetic field that is more tangled on smaller scalesmay be more difficult to amplify by shear amplification.The simulation set { Beta400,Tight,Loose,Tangential } ex-1 Fig. 12.—
Slices through the plasma β for the simulations withvarying initial configuration at the epoch t = 2.5 Gyr. Each panelis 500 kpc on a side. Major tick marks indicate 100 kpc distances. Fig. 13.—
Slices through the plasma β for the simulations withvarying initial configuration at the epoch t = 3.15 Gyr. Each panelis 500 kpc on a side. Major tick marks indicate 100 kpc distances. plores the effects of varying the initial magnetic field spa-tial configuration, with “Beta400” being the default con-figuration with k = 2 π/
43 kpc − , “Tight” correspond-ing to a higher k = 2 π/
15 kpc − , “Loose” correspondingto a lower k = 2 π/
120 kpc − , and “Tangential” corre-sponding to a tangentially oriented field (Equations 26-27). The initial average magnetic field strength for eachof these simulations is the same, corresponding to β =400. Figures 12 through 15 show slices of the plasma β through the center of the domain for the epochs t = 2.5,3.15, 3.75, and 4.5 Gyr after the beginning of the simu-lation. In contrast to our results for varying the initial β , there is not a similarly strong dependence of the evo-lution of the magnetic field on its initial configuration.About 2 Gyr after the initial disturbance, there is little Fig. 14.—
Slices through the plasma β for the simulations withvarying initial configuration at the epoch t = 3.75 Gyr. Each panelis 500 kpc on a side. Major tick marks indicate 100 kpc distances. Fig. 15.—
Slices through the plasma β for the simulations withvarying initial configuration at the epoch t = 4.5 Gyr. Each panelis 500 kpc on a side. Major tick marks indicate 100 kpc distances. qualitative difference between the runs. The field struc-ture becomes similarly tangled and amplified to similarvalues.Within the sloshing region, the magnetic field becomesextremely tangled with large fluctuations in strength.This is most apparent in the Tangential simulation,which begins with a relatively smooth tangentially ori-ented field but ends up with a randomly oriented fieldtangled on small scales within the sloshing region (seethe last panels of Figures 12-15). It appears that slosh-ing such as we are considering is more than enough todisrupt the tangential structure arising from the HBI,though future simulations including anisotropic heat con-duction will be required to confirm this (ZuHone et al.2011, in preparation).The field is most strongly amplified in the cold front2
Fig. 16.—
Initial (red lines) and final (at t = 5 Gyr, blue lines) profiles of the magnetic field strength for the simulations with varyinginitial β . Dot-dash lines indicate the final maximum and minimum values for each radial bin. Bin widths are 10 kpc. layers confining the sloshing region, but what is the over-all increase in magnetic energy due to this amplification?Figure 16 shows the radial profile of the magnetic fieldstrength, and Figure 17 shows the radial profile of theplasma β of the gas, both at the beginning and the endof the simulations with varying initial β . The final pro-files have a very similar shape across the simulations.Out to a radius of r ∼
100 kpc, the average magneticfield strength is roughly constant. There is an increase infield strength between radii r ∼ −
500 kpc (the radiiof the cold fronts at this point in the simulations), andafter this radius the field strength begins to decline withradius. The main difference between the profiles is theirmagnitude compared with the initial field strength. Forthe simulations with higher initial field strengths (sim-ulations
Beta100 and
Beta400 ), the final average fieldstrength within r ∼
100 kpc is lower than the initial av-erage field strength, though the field within the region ofthe cold fronts ( r ∼ −
500 kpc) is still stronger. Forthe simulations with lower initial field strength (simula-tions
Beta1600 and
Beta6400 ), we find that the average field is amplified at most radii from its initial value. Forthe simulations with varying initial spatial field config-urations for β = 400, we find that the final profiles arevery similar in magnitude and shape.Interestingly, the average field strengths for each sim-ulation with varying initial β , and especially for simula-tions Beta100 and
Beta400 (Figure 16 and Figure 17),are all similar within a factor of ∼ B ∼ a few µ G or β ∼ several hundred). Cold Front Morphology in the Presence ofMagnetic Fields
We find that magnetic fields alter the gas dynamics ofthe sloshing cold fronts. We begin by examining the sim-3
Fig. 17.—
Initial (red lines) and final (at t = 5 Gyr, blue lines) profiles of the plasma β for the simulations with varying initial β .Dot-dash lines indicate the final maximum and minimum values for each radial bin. Bin widths are 10 kpc. ulations that correspond to varying the initial plasma β .Figures 18 through 21 show temperature slices throughthe cluster center for the epochs t = 2.5, 3.15, 3.75, and4.5 Gyr after the beginning of the simulation for differ-ent initial β . Two effects are noticeable. The first isthat as the magnetic field strength is increased, the coldfronts appear more smooth and regular, even though theinstabilities are not suppressed completely. This is par-ticularly true in the case of the fronts with the small-est radii ( r ∼
50 kpc) at later times. These fronts aresmooth and well-defined in the simulations
Beta400 and
Beta100 , whereas they are more jagged and ill-defined inthe simulations
Beta6400 and
Beta1600 . The fronts atlarge radii ( r ∼ Beta100 ) are strongly affected by K-H in-stabilities (we will discuss the reason for this in Section4.2). The second significant effect is that as the mag-netic field strength is increased, the cool gas in the verycentral part of the core ( r ∼ <
50 kpc) is more resilient tothe effects of sloshing. We will examine this effect more quantitatively in Section 3.4.In contrast to the simulations where the initial mag-netic field strength was varied, the effect of varying theinitial magnetic field configuration is not as significant.Figure 22 shows temperature slices through the clustercenter for the epoch t = 4.5 Gyr after the beginning ofthe simulation. The smoothness of the fronts is aboutthe same for each of the simulations. The temperaturecontrasts across the fronts are also very similar, and thetemperature of the core ( r ∼ <
50 kpc) is approximatelythe same. In conformity with the results of the simula-tions with varying initial β , the cold fronts at large radiiare most susceptible to the onset of the K-H instability.One other particular aspect of the temperature mapsdeserves comment. Though in many cases, the front sur-faces are smooth due to the large magnetic field amplifi-cation in their surrounding layers, the effect of the mag-netic pressure can become dynamically important. Thisis particularly true in the case of the Beta100 simulation,as in many places in this simulation, the magnetic pres-4
Fig. 18.—
Slices through the gas temperature for the simulationswith varying initial β at the epoch t = 2.5 Gyr. Each panel is 500kpc on a side. Major tick marks indicate 100 kpc distances. Thecolor scale shows temperature in keV. Fig. 19.—
Slices through the gas temperature for the simulationswith varying initial β at the epoch t = 3.15 Gyr. Each panel is 500kpc on a side. Major tick marks indicate 100 kpc distances. Thecolor scale shows temperature in keV. sure is a sizeable fraction of, or even comparable to, thegas thermal pressure (i.e., β → ∼ −
30% in this simulation.
The Effect of Magnetic Fields on Mixing and CoreHeating Due to Sloshing
Sloshing motions bring higher-entropy cluster gas athigher radii into contact with the low-entropy gas of the
Fig. 20.—
Slices through the gas temperature for the simulationswith varying initial β at the epoch t = 3.75 Gyr. Each panel is 500kpc on a side. Major tick marks indicate 100 kpc distances. Thecolor scale shows temperature in keV. Fig. 21.—
Slices through the gas temperature for the simulationswith varying initial β at the epoch t = 4.5 Gyr. Each panel is 500kpc on a side. Major tick marks indicate 100 kpc distances. Thecolor scale shows temperature in keV. cool core, making it possible for these gases to mix andthe average entropy per particle of the core to increase(ZMJ10). We have shown in the previous paper that inthe absence of magnetic fields, the entropy of the corecan increase significantly from its initial state. The leftpanel of Figure 24 shows the evolution in average en-tropy within a radius r ≤
25 kpc (well within the coolingradius for a relaxed cluster) for each of our simulationswith varying β and the NoFields simulation, the latterusing the same hydrodynamical model as the invsicidsimulations of ZMJ10. The center is taken to be thecluster potential minimum, and “entropy” is defined as S ≡ k B T n − / e . Every simulation has an initial entropyper unit mass increase in the center associated with the5 Fig. 22.—
Slices through the gas temperature for the simulationswith varying initial configuration at the epoch t = 4.5 Gyr. Eachpanel is 500 kpc on a side. Major tick marks indicate 100 kpcdistances. The color scale shows temperature in keV. passage of the subcluster and the removal of low-entropygas from the cluster core. Following this initial increase,the evolution of the central entropy is strongly dependentof the details of the magnetic field. The right panel ofFigure 24 shows the final entropy profiles of these sim-ulations compared to the initial entropy profile. In the NoFields simulation, the cool core has been heated bysloshing and transformed to an isentropic core with a rel-atively high entropy. For the simulations with magneticfields, as the initial magnetic field strength is increased,the final entropy profile changes less; even though slosh-ing brings hot and cold gases in contact, the magneticfield suppresses their mixing (as did isotropic viscosityin ZMJ10).In contrast, we do not find that varying the initial mag-netic field spatial configuration changes the effectivenessof mixing. Figure 25 shows the evolution in average en-tropy within r ≤
25 kpc for simulations with differentinitial field configurations (all with initial β = 400), aswell as our control NoFields simulation. In each of themagnetized cases, the increase of entropy per unit massof the gas in the cluster core is suppressed (in comparisonto the
NoFields simulation, but the degree of this sup-pression is essentially independent of the initial field con-figuration. This is expected, given the weak dependenceof the final field configuration on the initial configurationthat we noted in Section 3.2In this analysis so far we have ignored the effects ofradiative cooling. In ZMJ10, a series of sloshing sim-ulations including radiative cooling were performed, todetermine if the mixing of gases due to sloshing providedsufficient heat to the cluster core to offset a cooling catas-trophe. It was found in that study that sloshing was ableto heat the core for a short period, but that this heatingwas insufficient to completely offset radiative cooling. Itwas also found that if the ICM is significantly viscous,mixing of hot and cold gases would be suppressed, andradiative cooling would be offset by a nearly insignificantamount of heating. Since we find that the effect of the magnetic field is to similarly suppress mixing and theheating due to mixing, we expect that a cooling catas-trophe would happen very quickly even in the presenceof sloshing if the ICM is magnetized.To confirm this, we have performed a simula-tion with radiative cooling included. The cool-ing function is calculated using the MEKAL model(Mewe, Kaastra, & Liedahl 1995), assuming the gas hasa uniform metallicity Z = 0 . Z ⊙ . Otherwise, the simula-tion is identical to the Beta400 simulation. We have fol-lowed the strategy of ZMJ10 in beginning the simulationshortly after core passage (corresponding to the epoch t = 2.0 Gyr in the non-radiative simulations), to avoid thecompression of the cluster core that occurs during thisperiod, which would lead to significant overcooling andhasten a cooling catastrophe. Our aim is to determinethe effect of sloshing on the cooling cluster core in iso-lation. To compare with this simulation, we have alsoran a simulation of an isolated, magnetized galaxy clus-ter with radiative cooling. We follow the simulations for2 Gyr.Figure 26 shows temperature slices at the epoch t =3.75 Gyr for the radiative and non-radiative sloshing sim-ulations. The most obvious difference between the twosimulations is the presence of the large temperature drop( T < r = 50 kpc for the radiative and non-radiative sloshingsimulations, compared to the radiative simulation of acluster in isolation. The entropy of the core in the ra-diative simulation drops very quickly, and is only mod-estly slowed down by the heat contribution from sloshingwhen compared to the case when there is no sloshing atall. This shows that the presence of the magnetic fieldsuppresses mixing of hot and cold gases to such a degreethat the effect of sloshing on the cooling rate of the coreis very modest. In line with the results from ZMJ10, wefind that the heat delivered from sloshing alone is unableto stave off a cooling catastrophe, especially in the caseof a magnetized ICM. However, thermal conduction hasnot been considered in these simulations. Since slosh-ing brings hot and cold gases into close contact, it mayyet result in significant heating of the cluster core due tothermal conduction. An investigation of core gas sloshingincluding the effects of anisotropic thermal conduction isthe subject of a future paper (ZuHone et al. 2011, inpreparation). Mock Observations
The observational signatures of sloshing cold fronts arethe spiral-shaped bright edges in the X-ray emission inthe cluster cores. To compare our simulations more di-rectly with these observations, we have constructed mockX-ray surface brightness observations from our simula-tion data. Each cell in the AMR grid has a photon emis-sion (in photons s − cm − ) given by ǫ γ = n e n H Λ γ ( T, Z ) (29)where n e and n H are the electron and hydrogendensities, respectively, and Λ γ ( T, Z ) is the emissiv-ity which depends on temperature and metallicity,6 × -10 × -10 × -10 × -10 -120 -100 -80 -60 -40 -20 0 P ( e r g c m - ) r (kpc) Thermal PressureThermal + Magnetic Pressure 0.004 0.006 0.008 0.01 0.012 0.014-120 -100 -80 -60 -40 -20 0 4 5 6 7 8 n e ( c m - ) T ( k e V ) r (kpc) DensityTemperature Fig. 23.—
An example of temperature and density fluctuations as a result of magnetic pressure from the
Beta100 simulation at theepoch t = 3.15 Gyr. Top panels: Gas density, temperature, and magnetic pressure slices through the center of the domain. Bottom panels:Profiles of the density, temperature, thermal pressure, and total (magnetic + thermal) pressure along the black lines in the top panels.
25 30 35 40 45 50 55 60 65 70 0 1 2 3 4 5 〈 S 〉 (r < k p c ) ( k e V c m ) t (Gyr)NoFieldsBeta6400Beta1600Beta400Beta100
10 100 1000 10 100 1000 S ( k e V c m ) r (kpc)OriginalNoFieldsBeta6400Beta1600Beta400Beta100 Fig. 24.—
The effect of magnetic fields of varying initial β on the entropy of the gas. Left: Evolution of the average entropy withina radius of 25 kpc for the simulations with varying initial β , compared to the simulation with no magnetic fields. Right: Final entropyprofiles (at t = 5 Gyr) for the simulations with varying initial β , compared to the simulation with no magnetic fields.
25 30 35 40 45 50 55 60 65 70 0 1 2 3 4 5 〈 S 〉 (r < k p c ) ( k e V c m ) t (Gyr)NoFieldsTightBeta400LooseTangential Fig. 25.—
Evolution of the average entropy within a radius of25 kpc for the simulations with varying initial field configuration,compared to the simulation with no magnetic fields. which are assumed constant over one FLASH cell size.The emissivity is calculated using the MEKAL model(Mewe, Kaastra, & Liedahl 1995), under the assumptionthat the cluster is situated at redshift z = 0 .
06 and a con-stant metallicity of Z = 0 . Z ⊙ , an assumption adequatefor our qualitative comparisons. Using this relation, thephoton luminosity of each sky pixel in photons s − in-tegrated over the chosen energy range (in this case, the Chandra band of 0.5-7.0 keV in the observer’s frame) andprojected along the line of sight is given by L γ = Z V ǫ γ dV ′ ≈ X i Λ γ,i n e,i n H,i ∆ V i (30)where the subscripts i refer to the quantities in eachAMR cell. The resulting surface brightness map is uni-formly gridded to the resolution of the smallest cell inthe simulation, ∆ x ∼ β forthe epochs t = 2.5, 3.15, 3.75, and 4.5 Gyr after the be-ginning of the simulation. There is a trend of increasingsmoothness of cold fronts with increasing magnetic field.Additionally, the simulations with higher magnetic fieldhave a brighter central core, in keeping with the resultfrom Section 3.4 that the cool core of the cluster is main-tained in these simulations. These results are similar tothe results that were obtained in ZMJ10 when viscositywas implemented. A more quantitative comparison withobservations will be given in a future paper. DISCUSSION
A Comparison of Amplified Field Strengths withEarlier Estimates
Under the assumptions of ideal MHD, the magneticfield lines are “frozen” into the flow, a condition thatcan be expressed by the equation (Chandrasekhar 1961): ∂ B ∂t + ∇ · ( vB − Bv ) = 0 (31)which, combined with the continuity equation, gives ddt (cid:18) B ρ (cid:19) = (cid:18) B ρ · ∇ (cid:19) v , (32) which implies that along shear flows the magnetic fieldwill be stretched and amplified. Keshet et al. (2010) de-rived an analytic estimate for the shear amplification ofthe magnetic field along cold fronts. From Equation 11from Keshet et al. (2010), we have (under the assump-tion of compressibility): B φ B r ′ ∼ t∂ r ′ v ∼ M i T / (cid:18) ∆10 kpc (cid:19) − (cid:18) t yr (cid:19) (33)where M i is the Mach number of the flow inside the front, T ≡ T /(4 keV) is the temperature inside the front, ∆is the thickness of the shear layer in kpc, and t is thedevelopment time of the magnetization layer. B φ is theamplified field strength just under and parallel to thefront, and B r ′ is the initial magnetic field perpendicu-lar to the front surface. To compare with the amplifi-cation of β seen in our simulations, we first note thatfor an initially random, tangled field the magnetic en-ergy in the component perpendicular to the front surfaceis roughly one-third of the magnetic energy in the totalfield, B r ′ / π ∼ (1 / B / π . For the final amplified field,we expect that B φ / π ∼ B / π . Setting B i = √ B r ′ and B f = B φ , assuming B f ≫ B i , and rearranging interms of β , we find: β f β i ∼ . M i T (cid:18) ∆10 kpc (cid:19) (cid:18) t yr (cid:19) − (34)where β i and β f are the initial and final plasma β . Tak-ing representative numbers from our simulations, we have M i ∼ . − . T ∼ ∼ −
10 kpc,and the amplified magnetic layers typically take a few × yr to develop. Under these conditions, the decreasein β should be β f /β i ∼ . − .
01, correspondng to amagnetic field energy amplification B f /B i ∼ − Smooth and Sharp Cold Fronts with MagneticFields
Our simulations demonstrate that for magnetic fieldstrengths compatible with those inferred from obser-vations, sloshing will result in shear amplification andstretching of the magnetic field lines along the cold fronts.The stretched and amplified fields will suppress instabil-ities of the fronts and help preserve the fronts’ smoothshapes. The efficacy of this is somewhat dependent onthe initial strength of the magnetic field, as stronger ini-tial fields are amplified more quickly to strengths thatcan suppress the instabilities.In addition to suppressing these instabilities, the mag-netic fields have another related effect which results inthe persistence of smooth and high-contrast cold fronts.By suppressing the mixing of high and low-entropy gases,the magnetic fields ensure that the densest gas in thecluster remains cold compared to the surrounding gas.Ghizzardi et al. (2010) pointed out a correlation betweenclusters with sloshing cold fronts and clusters with steepentropy gradients in the core. This was predicted inAM06; if the entropy gradient in the cluster core is notsignificant, the entropy contrast is not sufficient for the8
Fig. 26.—
Slices through the gas temperature for non-radiative and radiative simulations of gas sloshing at the epoch t = 3.0 Gyr. Thecolor scale shows the temperature in keV. Each panel is 500 kpc on a side. 〈 S 〉 (r < k p c ) ( k e V c m ) t (Gyr)Non-Radiative w/ SloshingRadiative w/ SloshingRadiative w/out Sloshing Fig. 27.—
Evolution of the average entropy within a radius of50 kpc for simulations with and without radiative cooling. cool gas pushed out of the disturbed cluster potentialminimum to flow back and for sloshing to begin. Simi-larly, sloshing will persist in generating cold fronts withhigher contrast if the steep entropy gradient is main-tained during the sloshing period. Since sloshing mixeshigh and low-entropy gas, possibly eliminating this gra-dient (ZMJ10), mixing should be suppressed in orderto maintain the sloshing. Our simulations indicate thatthe stronger the magnetic field is, the longer the originalsteep entropy gradient is maintained.In V01/V02, the lack of evidence for the growth of theKelvin-Helmholtz instability on the surface of a promi-nent cold front in the merging galaxy cluster A3667 wasused to argue for the existence of a magnetic field of B ∼ µ G parallel to the front. This was on the basis of asimple stability analysis for tangential perturbations ona shearing surface in the presence of a magnetic field. Itis instructive to see how the same analysis fares when ap- plied to our simulated cold fronts. In particular, can themagnitude of the actual magnetic field strength in oursimulation be predicted? For this purpose we have cho-sen a few fronts in our simulations and have carried outthe same rough calculation, made more straightforwardby the fact that we have direct access to the relevantquantities (density, velocity, etc.).The dispersion equation for small tangential-discontinuity perturbations in a perfectly con-ducting, incompressible plasma can be written as(Landau & Lifshitz 1960, in Gaussian units) ρ h ( ω − k ∆ v ) + ρ c ω = k (cid:18) B h π + B c π (cid:19) (35)where B h and B c are the magnetic field strengths in thehot and cold gases, respectively, ρ h and ρ c are the gasdensities in the hot and cold gases, v is the shear velocitydifference across the front, and ω and k are the pertur-bation frequency and wavenumber. The discontinuity isstable if (V01, V02, Keshet et al. 2010) B h π + B c π > ρ h ρ c ρ h + ρ c (∆ v ) (36)The treatment of the gas as incompressible, which sim-plifies the analysis, is justified by the fact that the Machnumbers of the gas flows in these regions are relativelylow ( M ∼ < . β , one that appears at the epoch t = 2.5 Gyrand another that appears at the epoch t = 3.15 Gyr.These fronts have been singled out due to their smoothappearance in the simulations with higher magnetic field,but the evidence of instability when the magnetic fieldstrength is lower. The two fronts and the front cross-sections along which we examine the gas quantities areshown in Figure 32. In the simulations Beta100 and
Beta400 , the chosen fronts are smooth and sharp, butin the simulations
Beta1600 and
Beta6400 the fronts arevisibly disturbed by instabilities.9
TABLE 2Cold Front Stabillity Analysis
Front ∆ v (km/s) ρ c /ρ h (10 − g cm − ) B pred ( µ G)1 250 4.0/1.5 13.12 150 6.5/3.5 8.0Front B sim ( µ G)Beta100 Beta400 Beta1600 Beta64001 19.4 11.2 5.0 1.12 15.9 7.1 5.0 3.2
Fig. 28.—
Projected X-ray brightness (in the 0.5-7.0 keV band)for the simulations with varying β at the epoch t = 2.5 Gyr. Thebrightness scale is square root and the same for each panel. Eachpanel is 500 kpc on a side. Fig. 29.—
Projected X-ray brightness (in the 0.5-7.0 keV band)for the simulations with varying β at the epoch t = 3.15 Gyr. Thebrightness scale is square root and the same for each panel. Eachpanel is 500 kpc on a side. Fig. 30.—
Projected X-ray brightness (in the 0.5-7.0 keV band)for the simulations with varying β at the epoch t = 3.75 Gyr. Thebrightness scale is square root and the same for each panel. Eachpanel is 500 kpc on a side. Fig. 31.—
Projected X-ray brightness (in the 0.5-7.0 keV band)for the simulations with varying β at the epoch t = 4.5 Gyr. Thebrightness scale is square root and the same for each panel. Eachpanel is 500 kpc on a side. β ,which is accurate to approximately 10% and is sufficientfor our current purpose. Table 2 shows the densities,shear velocities, and predicted minimum magnetic fieldstrengths for the two fronts if they are stable, and the ac-tual total magnetic field strengths in the four simulations Beta100,Beta400,Beta1600 and
Beta6400 .In the
Beta100 simulation, the magnetic fields at thefront surfaces are stronger than the minimum value re-quired for stability, consistent with the stability of thefronts. In the case of the
Beta400 simulation, the frontsare stable, while the actual magnetic field strengths arevery close to the values necessary to stabilize the front.In the
Beta1600 and
Beta6400 simulations, the frontsare not stable, and the corresponding magnetic fieldstrengths are far less than the values required for sta-bility.Strictly speaking, this analysis is only valid in theincompressible limit for a plane-parallel surface (seeChurazov & Inogamov 2004, for an alternative hypothe-sis for front stability based on the front curvature). Giventhe approximations of the above qualitative stability esti-mate, the predicted field estimates based on this stabilityanalysis agree quite well with the actual field strengthsthat are capable of stabilizing the fronts.
Amplification of Magnetic Fields by Sloshing:Implications
Magnetic fields amplified by sloshing motions may haveother observable effects. One such effect that has beensuggested by previous works is that of radio mini-halos.Mini-halos are regions of diffuse synchrotron emissionfound in the cooling core regions of some relaxed clus-ters. A possible connection to mini-halos and sloshingwas suggested by Mazzotta & Giacintucci (2008), whodiscovered a correlation between the radio minihalo emis-sion and the regions bounded by the sloshing cold frontsin two galaxy clusters. Other clusters with evidenceof sloshing cold fronts also sport minihalos, includingRXC J1504.1-0248 (Giacintucci et al. 2011) and Perseus(ZuHone et al. 2011, in preparation).The existence of radio mini-halos necessitates asource for the relativistic electrons. Two modelshave been proposed: the hadronic model (Dennison1980; Vestrand 1982; Blasi & Colafrancesco 1999;Dolag & Enßlin 2000; Pfrommer & Enßlin 2004) andthe reacceleration model (Cassano & Brunetti 2005;Cassano et al. 2007; Brunetti & Lazarian 2011). In thehadronic model, relativistic electrons are produced asbyproducts of hadronic interactions of cosmic ray pro-tons with thermal protons in the ICM, the resulting elec-trons producing synchrotron radio emission. The coin-cidence between the radio emission and the cold frontsin the hadronic model scenario would be due to theshear-amplified magnetic fields that are produced as a re-sult of the sloshing, as suggested by Keshet et al. (2010)and seen in our simulations. Alternatively, MHD tur-bulence driven by the sloshing motions may acceleraterelativistic electrons via damping of magnetosonic waves(Eilek 1979; Brunetti & Lazarian 2007). Numerical ex-periments to explore the relationship between gas slosh-ing in relaxed clusters and radio mini-halos is the subject of a forthcoming paper (ZuHone et al. 2011, in prepara-tion).Since the sloshing motions in our simulations createstrong magnetic fields ordered on large scales, it maybe possible to detect these fields via Faraday rotationmeasurements. A field line directed along our line ofsight will produce a rotation measure of the polarizedradio emission given byRM[rad m − ] = 812 Z L n e B k d l (37)where n e is the electron number density in cm − , B k is the parallel magnetic field strength in µ G, and L isthe length of the source in kpc along the line of sight.If the field lines in the amplified layers are partially di-rected along the line of sight, it should be possible todetect them in rotation measure observations of back-ground radio galaxies. In Figure 33, we give examplesof simulated RM maps from the Beta400 simulation atthree different epochs. In the early stages of sloshing(left and center panels), long, coherent structures in theRM map with large values are coincident with the placeswhere the cold fronts can be seen in X-rays. At latertimes, the layers are not as prominent, as the field am-plification is weaker and the field within the cold frontsis more tangled (right panel). The values of the rota-tion measure in these maps are comparable to those seenin cool-core clusters, such as Perseus (Taylor et al. 2006)and Hydra A (Taylor & Perley 1993). The spatial coinci-dence of a background radio galaxy with a cold front maybe a prime opportunity to find some observational con-firmation of a strong magnetic layer along a cold front.Alternatively, the CMB itself may be used as the sourceof polarized photons (Ohno et al. 2003), possibly allow-ing estimates of the magnetic field strength to be madeover a large spatial area of the cluster. Due to the veryspecific alignment of magnetic field layers with sloshingcold fronts, and the results of this work suggest that clus-ters with observed cold fronts would be good candidatesfor such a study.Caution should be taken, however, when interpret-ing these results. The accuracy of magnetic field es-timates from the RM method depends on the statisti-cal independence of the fluctuations in magnetic fieldand the thermal electron density. Anticorrelated fluctua-tions between the electron density and the magnetic fieldstrength will result in an underestimate of the magneticfield in these regions, since lower-density fluctuations willbe weighed less (Beck et al. 2003). This could be a pos-sible concern for detecting the magnetized layers seen inour simulations, as Figure 23 shows that they tend tobe associated with fluctuations of underdense gas. Thefluctuations that we see are on the order of ∼ SUMMARY
We used high-resolution magnetohydrodynamic simu-lations of gas sloshing in the cluster cool cores initiatedby the infall of subclusters. We study the effect of suchsloshing on the magnetic field in the intracluster medium1
Fig. 32.—
Fronts chosen for stability analysis. Plots are of gas density, where the scale is the same for all panels. The fronts chosen aremarked with a green line. Left set of panels: Front 1, for the simulations with varying β at the epoch t = 2.5 Gyr. Right set of panels:Front 2, for the epoch t = 3.15 Gyr. Each panel is 150 kpc on a side. The color scale is different from that in Figures 28 through 31. Fig. 33.—
Simulated rotation measure maps for selected epochs in the
Beta400 simulation. Each panel is 500 kpc on a side. Color scaleshows rotation measure in rad/m . and the effect of this field on the cold fronts seen in X-ray observations of relaxed galaxy clusters. We exploreda range of initial conditions for the magnetic field, includ-ing varying its initial strength and spatial configuration.Our results show that as a result of the shear flowsaccompanying the sloshing motions, the magnetic fieldenergy is increased significantly along the cold front sur-faces. The degree of amplification of the magnetic fieldstrength along these surfaces is up to an order of mag-nitude, resulting in an overall energy of the field B / π that is amplified by up to an order of magnitude or twofrom its average value prior to the onset of sloshing. Inparticular, in the layers along the cold fronts, fields withinitial strengths of 0.1-1 µ G may be amplified to tens of µ G. The final strength of the magnetic field is dependenton the initial field strengths, but weakly dependent onthe initial spatial configuration of the field. Our resultsfor the field strengths along the fronts are in agreementwith previous analytic estimates (Keshet et al. 2010).The sloshing motions result in a magnetic field that is tangled on small scales within the cluster cool core, andordered structures along the fronts, regardless of the ini-tial field configuration. Importantly, we find that re-gardless of the initial field strength or configuration, thefinal field strength averaged over the sloshing region isvery similar, implying the field strength does not increasewithout limit due to amplification by sloshing, but satu-rates.If the increase of the magnetic field strength is highenough, our simulations show that perturbations thatare able to grow due to the Kelvin-Helmholtz instabil-ity and disrupt the front surfaces are suppressed. Thedegree of suppression of these perturbations is highly de-pendent on the magnetic field strength of the cluster.Because the degree of amplification of the magnetic fieldis similar across the simulations, the simulations with thehighest initial magnetic field strength are most effectiveat maintaining smooth front surfaces. There is very littledependence of the effectiveness of the field to preserve thefronts on the initial spatial configuration of the magnetic2field, because the final field does not remember much ofits initial configuration. The cold fronts at high radiiare most susceptible to the onset of the K-H instability,mainly due to the weaker magnetic fields at those radii.The ordered magnetic fields also suppress mixing inthe intracluster medium. Sloshing in a cool-core clusterbrings hot and cold gas in close contact. In the absenceof these magnetic fields, these gases mix because of thedevelopment of instabilities, resulting in a net increasein entropy per gas particle in the core (ZMJ10). We findthat as the average magnetic field strength in the clus-ter is increased, the mixing of gases is inhibited, and thesloshing-induced heating of the core is hindered. Thesimulations with the strongest magnetic fields result invery little change to the radial entropy profile of the clus-ter. When cooling is included, we find that the magneticfields suppress mixing to such a degree that the heat con-tributed to the cluster core from sloshing is negligible.The interplay of sloshing motions and magnetic fieldsin relaxed clusters may have important consequences forfuture simulations and observations. The strong mag-netic fields, ordered on large scales, produced by sloshingcould potentially be detected in rotation measure maps,if a radio galaxy happens to be located behind a frontin the plane of the sky. Several examples exist of radiominihalos associated with X-ray cold fronts in relaxedclusters. Our simulations show that within the slosh-ing region, magnetic fields are amplified, which could be partially responsible for the correspondence between theradio emission and the X-ray fronts (Keshet et al. 2010).The sloshing motions are also likely to generate turbu-lence, and it is possible that this turbulence reacceleratesan existing population of relativistic electrons, which willbe addressed in a future paper (ZuHone et al. 2011, inpreparation).We would like to thank the anonymous referee for help-ful and constructive comments and suggestions. JAZthanks Ian Parrish, Bill Forman, Eric Hallman, and Ma-teusz Ruszkowski for useful discussions and advice. Cal-culations were performed using the computational re-sources of the Smithsonian Insitution’s Hendron DataCenter, Argonne National Laboratory, and the NationalInstitute for Computational Sciences at the Universityof Tennessee. Analysis of the simulation data was car-ried out using the AMR analysis and visualization toolsetyt (Turk et al. 2011), which is available for downloadat http://yt.enzotools.org , and for which MatthewTurk provided considerable help toward getting it work-ing for the analysis for this paper. JAZ is supported bythe NASA Postdoctoral Program. The software used inthis work was in part developed by the DOE-supportedASC / Alliances Center for Astrophysical ThermonuclearFlashes at the University of Chicago.
APPENDIX A. RELAXATION OF THE MAGNETIC FIELD
The initial magnetic fields in our simulations are not in an equilibrium configuration. On the one hand, the fieldsthemselves are too dynamically weak to affect significantly the density and temperature structure of the gas, so we canassume that this structure will not change significantly until sloshing begins. However, if left to itself, the magneticfield of the cluster will rearrange itself to an equilibrium configuration, which may be a different configuration fromwhich the simulation was started. In our default simulation setup, this process of relaxation is occuring while thesubcluster is approaching the main cluster core and at the very beginning of the sloshing period. Since this period isnot long, the relaxation is incomplete at the time of the passage of the subcluster, when the effects brought on by theencounter become much more important. To investigate the effects of this relaxation of the field on our results, we haverun a second simulation with the same parameters as
Beta100 , except that the cluster has been allowed to evolve inisolation for 6 Gyr, when the magnetic field has approached an equilibrium configuration. Since this is the simulationwith the strongest magnetic fields, we expect that the differences between the relaxed version and the original versionwill be most pronounced. We then begin the evolution of the simulation in the same fashion as the
Beta100 simulation.Figure 34 shows the radial profile of the plasma β at the end of the relaxation of the single-cluster, compared withthe radial profile of the plasma β before any evolution. The final profile runs from β ∼
250 in the center of the clusterto ∼
400 in the outskirts of the cluster. We might expect, based on our results from the simulations with varyinginitial β , that the effects caused by the evolution of this initial field due to sloshing would be intermediate between the Beta100 and
Beta400 simulations. Figure 35 shows a comparison between temperature slices of the relaxed simulationand the
Beta100 simulation. In the relaxed-field simulation, the Kelvin-Helmholtz instabilities are still suppressed, butnot as strongly as they are in the unrelaxed simulation. Indeed, the t = 3.75 Gyr snapshot of the relaxed simulationin Figure 35 looks very similar to the one for the Beta400 simulation shown in Figure 20. Finally, Figure 36 shows acomparison between the evolution of the average entropy of the cool core within a radius of 25 kpc vs. time betweenthe relaxed-field and
Beta100 simulations. We have also included the same evolution from the
Beta400 simulation forcomparison. The increase in the entropy of the core due to mixing in the relaxed simulation is intermediate betweenthe increase of the
Beta100 and
Beta400 simulations.We conclude from the results of this test that the primary differences between our
Beta100 simulation and its“relaxed” version are that the reduced magnetic field strengths present in the latter inhibit the growth of perturbationson the fronts less than in the former, and that mixing and the increase of entropy in the core are similarly less inhibited.This is in line with the conclusions from our simulations with varying initial β . B. RESOLUTION TEST
To test the robustness of our conclusions against the effects of varying resolution, we have performed a simulationwith the same relaxed-field, β = 100 setup described in the previous section, but with a finest cell size of ∆ x ∼
50 100 150 200 250 300 350 400 450 0 200 400 600 800 1000 β r (kpc) Beta100Relaxed Fig. 34.—
Radial profiles of the plasma β for the beginning of the relaxed simulation and the Beta100 simulation. half the cell size of the set of simulations described in this work. Increasing the resolution lowers the effective numericalviscosity of the simulation, allowing for smaller perturbations to be resolved and grow along the front surfaces. It alsoresolves smaller-scale fluctuations in the magnetic field, making it more difficult to maintain a coherent tangential fieldstructure across the front surfaces. Both effects may affect our conclusions.Figures 37 through 39 show slices through the temperature (in keV) and the plasma β parameter for the epochs t = 2.5, 3.15, 3.75, and 4.5 Gyr after the beginning of the simulation for the relaxed-field initial setup with resolutionsof ∆ x ∼ x ∼ Fig. 35.—
Slices through the gas temperature in keV for the
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