Three-Dimensional Simulations of Bi-Directed Magnetohydrodynamic Jets Interacting with Cluster Environments

2.2 MB·English

by  S. M. O'Neill

Accepted, to appear in ApJ, v710, 2010 February
PreprinttypesetusingLATEXstyleemulateapjv.11/10/09
THREE-DIMENSIONAL SIMULATIONS OF BI-DIRECTED MAGNETOHYDRODYNAMIC JETS
INTERACTING WITH CLUSTER ENVIRONMENTS
S. M. O’Neill1, T. W. Jones2
Accepted, to appear in ApJ, v710, 2010 February
ABSTRACT
We report on a series of three-dimensional magnetohydrodynamic simulations of active galactic
nucleus (AGN) jet propagation in realistic models of magnetized galaxy clusters. We are primarily
interested in the details of energy transfer between jets and the intracluster medium (ICM) to help
0 clarify what role such flows could have in the reheating of cluster cores. Our simulated jets feature a
1
range of intermittency behaviors, including intermittent jets that periodically switch on and off and
0
onemodeljetthatshutsdowncompletely,naturallycreatingarelicplume. TheICMintowhichthese
2
jets propagate incorporates tangled magnetic field geometries and density substructure designed to

n mimic some likely features of real galaxy clusters. We find that our jets are characteristically at least
a 60% efficient at transferring thermal energy to the ICM. Irreversible heat energy is not uniformly
J distributed, however, instead residing preferentially in regions very near the jet/cocoon boundaries.

1 While intermittency affects the details of how, when, and where this energy is deposited, all of our
1 models generically fail to heat the cluster cores uniformly. Both the detailed density structure and
nominally weak magnetic fields in the ICM play interesting roles in perturbing the flows, particularly

] when the jets are non-steady. Still, this perturbation is never sufficient to isotropize the jet energy
O
deposition,suggestingthatsomeotheringredientisrequiredforAGNjetstosuccessfullyreheatcluster
C cores.
. Subject headings: galaxies: jets – galaxies: clusters: general – methods: numerical – MHD
h
p
- 1. INTRODUCTION historyofrealisticjetlifetimesanddutycycles. Someat-
o
temptstoestimateAGNintermittencyarebasedlargely
r Over the past decade, X-ray observations of galaxy
t on energetics. Soker et al. (2001), for example, estimate
s clustershaveclearlyillustratedtheimmenseenergiesas-
a sociated with interactions between active galactic nuclei jet lifetimes of 107 −108 years with duty cycles of 5%
[ (AGNs) and their environments. For example, discover- or less to stifle cooling while matching optical and ra-

iesofX-raycavitiesassociatedwithAGNradio-emitting dio observations of cooling cores. Observations of bub-
1
‘relic bubbles’ (e.g. McNamara et al. 2000; Fabian et al. bles in the absence of AGN (Mazzotta et al. 2002) have
v
7 2000; Fujita et al. 2002; Nulsen et al. 2002; Wise et al. also been used to estimate lifetimes of 107 −108 years
4 2007; Sanders et al. 2009) demonstrate that radio lobes and duty cycles of at most 10%. Other approaches focus
7 associated with current or previous AGN activity often more on timescales associated with observed features in
1 completelydisplacethesurroundingintraclustermedium individualsources, suchasthoseconductedbyFabianet
. (ICM). Likewise, X-ray observations of ‘ghost cavities’ al. (2003), Forman et al. (2005), Jamrozy et al. (2007),
1
devoid of radio emission (e.g. McNamara et al. 2001; and Blanton et al. (2009), all of whom find a rate of
0
Mazzotta et al. 2002) have shown that these cavities one event per 107 −108 years. Observations of the so-
0
maintain some structural integrity even after the radio called double-double radio sources (Schoenmakers et al.
1
population has aged beyond the limits of detectability. 2000), in which two distinct episodes of jet activity are
:
v The energies associated with these plasma bubbles are clearly visible, have implied rather short AGN lifetimes
i foundtobeupwardof1059−1060 erg(Bˆırzanetal.2004; of ∼106 years at most. Croom et al. (2004) take a com-
X
Dunnet al.2005;Wiseet al.2007), suggestingthatthey pletely different approach, using QSO clustering to esti-
r could play a significant role in stifling cooling flows and mateAGNintermittency, andtheirresultsuggestsAGN
a
maintaining the ∼ 2 keV temperature floor observed in lifetimes of 106−107 years. Most recently, Bˆırzan et al.
clusters (e.g. Peterson et al. 2001; Fabian et al. 2001; (2009)useacompletesampleofclusterstoestimateduty
Kaastraetal.2001;Tamuraetal.2001). Thisideaisfur- cycles of at least 60% for AGN outbursts in cooling flow
ther supported by simulations of cluster formation and clusters. Diehl et al. (2008) also examine a large sample
evolution with AGN feedback, such as those conducted ofclustersandinferthattheobservedsizedistributionof
by Bru¨ggen et al. (2005) and Sijacki & Springel (2006). X-ray cavities is most consistent with either continuous
Observations of detached relic bubbles and evolved hydrodynamicinflationordiscreteproductionofbubbles
ghost cavities also suggest that a complete picture of by magnetically dominated jets.
AGN interactions with the ICM should incorporate a Although there is much observational evidence to sup-
port models of intermittent jets that feed energetic
1UniversityofMaryland,DepartmentofAstronomyandMary- plasma bubbles to the ICM, much still needs to be un-
land Astronomy Center for Theory and Computation, College derstoodabouttheprecisenatureofenergyflowinthese
Park,MD,20742;[email protected]
systems. Simple analytic models, such as those devel-
2School of Physics and Astronomy, University of Minnesota,
Minneapolis,MN55455;[email protected] oped by Cioffi & Blondin (1992), Kaiser & Alexander
2
(1997), and Komissarov & Falle (1998), are powerful unclear how this approach will affect the case of magne-
tools for analyzing the basic dynamics and energetics in tizedbubbles. Heinz&Churazov(2005), takingadiffer-
thesesystems,butarenotdesignedtoaddressthedetails ent approach, showed that hydrodynamic bubbles could
of energy transfer to the ICM. Furthermore, these mod- explicitly transform wave energy into heat that could be
els tend to feature arguments based upon self-similarity dissipated.
or constant jet luminosities, and so may not apply to re- Several groups have also explored the effects of jet
alistic ICM atmospheres or intermittent jets. Likewise, intermittency on flow dynamics and energetics. The
numerical studies that focused on jet dynamics in both earliest example of a simulated restarting jet was con-
the non-relativistic (e.g. Norman et al. 1982; Hardee & ducted by Clarke & Burns (1991), who found that sub-
Clarke 1992; Norman 1996; Carvalho & O’Dea 2002a,b; sequentjetsmovedquicklythroughpreviousjetmaterial,
Krause 2003, 2005) and relativistic regimes (e.g. Dun- forming a new bow shock in the process. Using a two-
can & Hughes 1994; Marti et al. 1997; Aloy et al. 1999; dimensional hydrodynamics code, Reynolds et al. (2002)
Mizuno et al. 2007; Keppens et al. 2008; Mignone et al. found that plumes from dead radio galaxies lifted sub-
2009) made great strides toward understanding how jets stantialamountsofICMmaterialastheyroseandthata
propagate, but were not necessarily interested in the de- largefractionofthetotaljetenergywentintoheatingthe
tails of energy transfer to the ICM. Rosen et al. (1999) ICM. This was followed by Basson & Alexander (2003),
provide a useful analysis of the primary differences be- who demonstrated in three dimensions that a buoyant
tweenrelativisticandnon-relativisticsimulations,point- jet-blown bubble continues to dredge up material from
ing out that jets of equal thrust have similar ages and the cluster core for much longer than the jet lifetime.
efficiencies in both cases, although relativistic jets typi- Alsointhreedimensions,Ommaet al.(2004)foundthat
cally develop smaller cocoons than their non-relativistic mostofthejetenergyinanoutburstendsuptoofaraway
counterparts. from the cluster core. In a subsequent simulation that
Recentnumericalstudiesofjetenergetics,ontheother includedcooling,Omma&Binney(2004)foundthatthe
hand, have contributed greatly to the understanding of amountofdense, coolgasimmediatelyabovethecentral
energy flow from AGN to their environments. Zanni et AGN had a pronounced effect on where most of the jet
al. (2005), for example, conducted an ensemble of two- energy was dissipated. While the previously mentioned
dimensional simulations of jets in detailed cluster envi- workbyVernaleo&Reynolds(2006)showedthatthisef-
ronments,findingthatasignificantfractionofjetenergy fect was never sufficient to completely offset cooling in a
isdepositedirreversiblyintotheICM.Likewise,ourpre- simple ICM atmosphere, recent simulations by Bru¨ggen
vious study of three-dimensional magnetohydrodynamic & Scannapieco (2009) suggest that the modeling of sub-
(MHD)jets(O’Neilletal.2005)foundthatenergytrans- grid turbulence may alleviate some of these problems.
fer to the ICM was efficient for both uniform and strati- Ourstudycombinesseveralofthesedifferentinvestiga-
fiedenvironments. Continuingworkinthreedimensions, tions,exploringtheinteractionsofintermittentjetswith
Vernaleo & Reynolds (2006) pointed out that simple hy- theICM.Weseektodeterminehowintermittencyaffects
drodynamic models of jet feedback characteristically de- energyflow,askinginwhatform,where,andhowrapidly
velop a relatively lossless low-density channel and sub- energyistransferredfromjet-producedstructurestothe
sequently fail to dissipate and distribute energy widely ICM. These simulations specifically examine whether a
enough to prevent catastrophic cooling. Heinz et al. combination of jet intermittency, magnetic stresses, and
(2006), however, attempted to address this issue by sim- ambient density fluctuations alone can lead to broader
ulating wobbling hydrodynamic jets in an evolving clus- distribution and dissipation of jet energy, or whether
ter atmosphere that served to isotropize the deposited something like the more chaotic jet model of Heinz et
energy. Whether the jet and jet-blown cocoon remain al. (2006), or a very high Reynolds number turbulence
characteristically stable or break apart or mix with the suchasthatproposedbyBru¨ggen&Scannapieco(2009),
ambientmedium(asillustratedrecentlyinGaibleretal. is required. Furthermore, we seek to understand the ba-
2009, for example) clearly influences the efficiency with sic dynamics and morphology of intermittent flows. We
which jets deliver energy to cluster cores. test how flow structures produced by intermittent and
Somerecentnumericalstudiesofrelicbubblepropaga- extinct jets evolve, how they transport and transfer en-
tion also have had implications for energy transfer and ergy,andhowtheyinteractwithandpotentiallyenhance
transport in the ICM. Simulations conducted by Robin- ambient magnetic fields.
son et al. (2004) and Jones & DeYoung (2005) in two- To address these questions, we employ a set of MHD
dimensions and Ruszkowski et al. (2007), O’Neill et al. simulations of jets in realistic cluster environments. The
(2009), and Dong & Stone (2009) in three dimensions, jets are oppositely directed, launching into an ICM with
for example, illustrated that rising MHD bubbles lift a full three-dimensional gravity, tangled magnetic fields,
great deal of ICM material in their wake. This material and density fluctuations designed to mimic cluster sub-
has the potential to mix and distribute energy within structure. Section2describesthedetailsofthenumerical
theICM.ArecentpaperbyBru¨ggenet al.(2009)points methods and the models employed. Section 3 contains a
out that this mixing at the bubble interface is further discussion of the results and astrophysical implications,
enhanced in the hydrodynamic case when turbulence is while Section 4 lists the main conclusions of this work.
treated with a subgrid approach, which explicitly mod-
els the energetics and sizes of turbulent eddies that are 2. CALCULATIONDETAILS
not properly resolved by the numerical grid. This is in Our simulations employ a second-order Eulerian total
contrast to the usual method of treating turbulence in variation diminishing (TVD) non-relativistic ideal MHD
which the computational grid resolution sets the effec- code, described in Ryu & Jones (1995) and Ryu et al.
tive scale of dissipation in the system. It is currently (1998). The code explicitly enforces the divergence-free
3
absence of a jet over three grid sound-crossing times, a
TABLE 1
time much longer than any of our jet simulations.
Summary of Simulations of Double Jets in
Galaxy Clusters A passive color tracer Cj, representing the mass frac-
tion of jet material, is introduced through jet flow to
ID1 JetIntermittency2 FinalAge identify material that has passed through the jet ori-
fice. Additionally, we include a population of passive,
nonthermal, relativistic cosmic ray particles in our mod-
ST Steadyinflowforduration ≈59Myr
I13 Switchon/offevery13Myr ≈173Myr els. Thesecosmicraysaretransported,injected,acceler-
I26 Switchon/offevery26Myr ≈104Myr ated, and aged in a self-consistent fashion to enable the
RE Onfor26Myr,thenoff ≈144Myr construction of realistic synthetic observations (see, e.g.,
AM Nojetsincluded ≈183Myr
Jones et al. 1999, Tregillis et al. 2001a, Tregillis et al.
2004, Mendygral et al. in prep).
1 All models feature identical computational grids Here, we describe five simulations in which the initial
(600kpc×480kpc×480kpc) and ICM structures.
ICM model features core density ρ0 = 8.33×10−26 ambient conditions are precisely the same, but the time
g cm−3 and pressure P0 = 4.0×10−10 dynecm−2, historiesofjetactivityvary. Inthefollowingtwosubsec-
correspondingtoacoresoundspeedc0=895kms−1 tions, we describe the jets and ambient media in detail.
andtemperature2.5keV. Ambientmagneticfieldsare Thephysicalparametersofeachsimulationaredescribed
t2anIgnlefdulwlyithacaticvhearsatcatteersi,stjiectcsorfeeavtaulrueeMB0ac∼h1n0umµGbe.r in Table 1.
Mj =30,correspondingtovelocityvj =0.0895c. Jet
densities are calculated from fixed ρj = η ρ0, with 2.1. Bi-directed Jets
η =0.01. Jet magnetic fields are completely toroidal
withmaximumstrengthBj =10µG. Jet inflow is introduced entirely along the ±x-axis
without any initial transverse component. The previ-
conditionformagneticfieldsthroughaconstrainedtrans-
ouslymentionedinternaljetboundaryconsistsofacylin-
port scheme detailed in Ryu et al. (1998). The numer-
dricalregionoriginatingfromx=0andextending6kpc
ical method conserves mass, momentum, and energy to
along±x. Theresultingjetshaveuniformcoresofradius
machine accuracy. Gravitational energy is handled in a
r =3 kpc, surrounded by a concentric transition annu-
fullyconservativefashionbyincludingthiscomponentin j
lus that smoothly connects to the ambient conditions at
the total energy of the gas. Additionally, energy fluxes
a radius of 7 kpc. The nominal core density contrast is
across the boundaries and changes in energy on the grid
η =ρ /ρ =0.01, and the jets initially enter in pressure
are tracked and output for each energy type to facilitate j a
equilibrium with their surroundings. The jet core mag-
analysis of energy flow (see Section 3.1). Model param-
netic field in this set of models is completely toroidal, of
eters (discussed in Section 2.2) are selected such that
the form B = B (r/r ) within the jet core, but that
the cluster cooling times are longer than the simulation φ 0 j
decreases quadratically to zero outside of the core re-
times (t >250 Myr in the cluster cores), allowing us
cool gion. This jet field was chosen to be strictly toroidal so
to neglect cooling in our simulations.
thatitcouldbeinitiallydivorcedfromtheambientfield.
All simulations described here are performed on a
The fiducial ratio of thermal pressure to magnetic pres-
three-dimensional Cartesian grid of full physical dimen-
sure in the jet (measured at the boundary of the core)
sions x = 600 kpc, y = z = 480 kpc. Each zone repre-
is β ∼ 100, and the associated (maximum) inflowing jet
sents one cubic kiloparsec of volume with ∆x = ∆y =
field strength is B =10 µG.
∆z = 1 kpc. Given that the total jet diameter is 14 0
For this set of models, the jet luminosity can be ex-
kpc (see Section 2.1), this is sufficient to resolve shocks
pressed in terms of the steady jet values of the density
and multiple scales of turbulence within the jet and co-
ρ , velocity v , and pressure p as
coon. With the exception of one model (AM, described j j j
in Section 2.1), these simulations feature sets of oppo-
sitely directed jets emanating from a cylindrical internal (cid:34) (cid:18) ρ (cid:19)(cid:18) v (cid:19)3 (cid:18) p (cid:19)(cid:18) v (cid:19)(cid:35)
boundary located in the central region of the computa- Lj ≈2.12×1045 1.90 ρ v + p v erg s−1,
tional grid (x = y = z = 0). The jet region is updated j j j j
prior to each TVD step, and the settings in this region (1)
are used to control jet luminosity and intermittency. ignoring the relatively minor contributions of magnetic
Sincethejetsarelaunchedfromthecenterofthecom- andgravitationalenergytothejetluminosity. Sinceitis
putationalgrid,theoutergridboundaryconditionshave thevelocity(andnottheluminosity)thatgoessmoothly
little influence on the evolution of jet/cocoon structures. to zero in the transition annulus, the size of the annu-
Still, the outer boundaries are crucial in maintaining lus enters into the kinetic and thermal energy flux terms
the dynamical stability of the constantly evolving am- differently, resulting in the prefactor of 1.9 (see the Ap-
bient medium. In the outer grid boundaries, we use a pendixfordetails). Thisanalyticexpressionismorethan
set of modified continuous boundaries designed to main- 99% accurate over the lifetime of a steady jet. The first
tainapproximatehydrostaticequilibriumintheambient jet model features two steady Mach 3 jets (labeled ’ST’
mediumwhileminimizingtheimpactofsmall-amplitude for ’steady’) that correspond to a physical flow speed
waves incident upon the boundaries. Specifically, we ap- v ∼ 0.1c and jet luminosity L = 6.1 × 1045 erg s−1
j j
plytheconstraintsthatthesoundspeedbeconstantand per jet. The other runs in this series of simulations fea-
thathydrostaticequilibriumapplytosetthedensityand ture jets with variable activity. In all cases, the maxi-
pressure conditions across a given boundary. We find mum injected jet speed and Mach number match those
that these boundaries maintain the total mass and en- of the ST model, and it is worth noting explicitly that
ergyofthegridtoanaccuracygreaterthan99.5%inthe these assumed jet luminosities and speeds (correspond-
4
Fig. 1.— Volume rendering of flow speed for theST (upper left), I26 (upper right), I13 (lower left), and RE (lower right) models, after
the jet disturbances have nearly reached the computational grid boundary. The contrast in the RE image has been enhanced to increase
visibility. Animationsofthesequantitiesasseenfromseveraldifferentangleswillbeavailablethroughtheelectronicversionofthispaper.
ing to a Lorentz factor of 1.004) are consistent with a model (labeled ’AM’ for ’ambient’) in which the jets are
non-relativistictreatment. Intworuns,thejetsintermit- never activated. This final model allows us to separate
tently switch on or off at regular intervals. In these two theevolutionoftheambientmediumfromjet-drivenbe-
cases, the jets switch states every 26 Myr (labeled ’I26’) haviors to correctly assess the role of the environment in
and 13 Myr (labeled ’I13’), respectively. This switch these simulations.
is accomplished by ramping the density, pressure, and Figure1showsvolumerenderingsofflowspeedatlate
momentum density exponentially over a 3.27 Myr (1.64 times for each of the four jet-driven models. In each
Myr) timescale for the I26 (I13) run. The exponential frame, the jets originate near the center of the image at
ramp changes the density and pressure to a volume- an angle with respect to the viewer, with the approach-
averaged value sampled from a spherical region imme- ing jet pointing toward the lower right and the receding
diately surrounding the jet, allowing for a smooth tran- toward the upper left. All four of the jet-driven simula-
sition from active jets to a quiescent state. In the fourth tions run until wave or shock disturbances from the jets
model (labeled ’RE’ for relic), designed to mimic radio reachagridboundary. Thistimescalevariesfrommodel
relic sources, the jets are completely shut down after 26 to model depending on how the flow is driven, and the
Myr and are not restarted. Additionally, there is a fifth total simulation times are given in Table 1.
5
Fig. 3.— Initial tangled ICM magnetic field lines. Arrows on
field lines indicate direction while intensity reflects field strength
(light: stronger, dark: weaker). Jets emerge from the region near
thecenteroftheimage.
which generates a gravitational acceleration:
Fig. 2.—ICMdensity,pressure,andtemperatureprofiles. Values
orefpρr0es,ePnt0,thaendlocTa0tiaornesgoifvtehneiInCMTabdleens1i.tyTcohreelraabdeilis. rc1 and rc2 g(r)=−4πGrd3mρs (cid:20)ln(cid:18)1+ r (cid:19)− r (cid:21) (4)
r2 r r+r
dm dm
2.2. Galaxy Cluster Environments
where r = 400 kpc is taken to be the characteristic
In all five models, we employ a single type of ambient dm
dark matter core radius. Another free parameter is the
medium designed to incorporate the essential properties
dark matter density, ρ . Rather than deriving this from
of galaxy cluster environments. The gas density base s
estimated cluster masses, we instead selected the value
profile is a double-β model (e.g., Ota & Mitsuda 2004),
that would produce a reasonable temperature distribu-
similartothosetypicallyusedtoobservationallydescribe
tion (see below). In this case, the temperature profile
clusters, of the following form (see Figure 2):
selected corresponds to ρ ≈ 4.3×10−26 g cm−3. For a
s
cluster with a virial radius r = 2 Mpc, this leads to a
v
ρ (r)=ρ (cid:34) f1 + f2 (cid:35), (2) vofiraiaflemwaosfstMhevm=as5s×of1t0h1e4PMe(cid:12)rs,ewushciclhusitserw(iet.hgi.nEattfoarcitoert
a 0 (1+(rrc1)2)32β (1+(rrc2)2)32β al. 1998).
The pressure profile is determined by the requirement
where ρ =8.33×10−26 g cm−3 is the density at r =0.
0 thatthebaseatmosphere(i.e., ignoringthedensityfluc-
This profile features two density “cores” weighted differ-
tuations) be in hydrostatic equilibrium. The gravita-
ently: r =50kpcwithaweightf =0.9andr =200
c1 1 c2 tionalaccelerationandgasdensityprofilearesufficiently
kpc with a weight f = 0.1. These values were chosen
2 complex that the pressure profile is derived numerically
to be consistent with the best-fit values found through a
and is shown in Figure 2. The core value of the pres-
statistical analysis conducted by Ota & Mitsuda (2004) sure is P = 4.0×10−10 dyne cm−2. The pressure and
on 79 X-ray clusters. For these models, β = 0.7, which 0
density profiles can be combined to give a temperature
generatesanasymptoticprofileonlyslightlysteeperthan
profile (Figure 2), which is used as mentioned to provide
1/r2whenr >>r >r . Ontopofthisprofile,wehave
c2 c1 the normalization of gravity. The temperature profile
superposed a Kolmogorov (1941) spectrum of density
was selected to resemble typical clusters (e.g., Vikhlinin
fluctuations with a maximum amplitude of ±0.10ρ (r)
a et al. 2006). As Figure 2 shows, the atmosphere has a
locally. This amplitude of adiabatic fluctuations in den-
coretemperatureT ≈2.5keV(assumingameanmolec-
sitycorrespondstotherangeofpressurefluctuationsob- c
ular weight µ = 0.5), with a total temperature range
served by Schuecker et al. (2004) in the Coma cluster.
of ∼ ±0.2 T over the computational grid. The sound
This initial spectrum of fluctuations is spatially tapered c
speed, with a profile identical to that of the tempera-
by a Gaussian envelope near the edges of the compu-
ture, features a core value c =895 km s−1.
tational grid to impede the development of unphysical 0
The ambient magnetic field (see Figure 3) is designed
flows at the grid edges.
to be a tangled, yet analytically specifiable, structure of
Forthesesimulations,gravityisderivedfromanNFW
the following form:
profile (Navarro et al. 1997), intended to model the un-
derlying cluster dark matter distribution:
B(cid:126) =B θˆ+B φˆ, (5)
θ φ
ρ
ρ = s , (3) where the two components are
dm ( r )(1+ r )2
rdm rdm
6
Fig. 4.— Average values of 100α ≡ 100PB/Pgas (solid line), Fig. 5.—EnergyflowintheSTmodel. Upperleft: acomparison
plottedasafunctionofradiusfromtheclustercenter. Thedotted of the known added kinetic energy (dashed line) to the measured
line,shownforcomparison,illustrates100α=1,correspondingto change (relative to the initial value) in kinetic energy on the grid
β≈100. (dotted line) as they vary in time. The total (kinetic + thermal
+ magnetic+gravitational) inflow energy is shown (solid line) as
a reference. Upper right: same for the thermal energy. Lower
left: same for the magnetic energy. Lower right: same for the
F (r)·m
B = 1 sinθcos(mφ) (6) gravitationalenergy.
θ r
and the measured change in energy on the grid for a given
energy type, we can determine how energy is converted
as the flow evolves. Additionally, we can use the passive
F (r)·n F (r)
B = 2 sin(nθ)− 1 sin(mφ)sin(2θ). (7) color tracer Cj to indicate what fraction of this added
φ r r energy enters the ICM. We are particularly interested
here in the amount of thermal energy transferred to the
The constants m and n are chosen to allow the field
ICM since this energy is most likely to be of relevance
to vary in φ and θ. In our simulations m = n = 3,
to the quenching of cooling flows. Again, we neglect ra-
to allow for several full field reversals over either an-
diativecooling;ourprimarygoalistounderstandenergy
gle. F (r) and F (r) are, in general, arbitrary functions
1 2
transfer from the jets.
of r that we use to approximate a constant β atmo-
Figures 5 - 8 illustrate the energy flows for each of
sphere with fluctuations. In practice, we choose F (r)=
1
the four jet simulation models we use. In each figure,
rB (r)[1+sin(a r)]andF (r)=rB (r)[1+sin(a r)]
a kr1 2 a kr2
four plots are shown, corresponding to the four types of
to provide field variation with radius. The constants
a = 2π/100 kpc−1 and a = 2π/58 kpc−1 are used energy present: kinetic, thermal, magnetic, and gravi-
kr1 kr2 tational. In each plot, the solid line represents the to-
toprovidefieldsthatvaryonspatialscalescomparableto
tal amount of energy added to the grid by the jets, mi-
or slightly larger than the width of the jet. These scales
nustheenergymeasuredtohaveexitedthegridthrough
shouldbeinterpretedasthemaximumlengthscalesover
the outer boundaries. In these models, the latter quan-
which the field varies in our clusters rather than the
tity is very small since the systems would be in hydro-
scales on which magnetic power is concentrated. The
static equilibrium were it not for the evolution of the
strongestfieldsinourclustersvaryoverafewtensofkilo-
added density perturbations and magnetic field fluctua-
parsecs,closertothevaluesinferredforobservedclusters
tions. In practice, the amount of energy flux across the
in Vogt & Enßlin (2005), for example. B (r) is simply
a outer boundaries is further reduced by the fact that the
the ambient field strength as derived from the pressure
density, pressure, and magnetic field are at their weak-
profile and the requirement that β ≈ 100 over a radial
est in these regions. The dashed line in each plot shows
average, as shown in Figure 4.
the integrated jet power for that particular energy type.
The dotted line shows the measured change in that en-
3. DISCUSSION
ergy type on the computational grid, excluding the jet
3.1. Energy Flows
inflow region that we treat as a boundary.
We begin our discussion with an examination of en- Since the systems were not in perfect equilibrium ini-
ergyflowsinthesesystems. Bycarefullytrackingthejet tially, the ambient medium relaxes as the simulations
power over time and comparing the integrated power to progress, causing some exchange of energy independent
7
Fig. 6.—SameasFigure5,butfortheI26model. Fig. 8.—SameasFigure5,butfortheREmodel.
Fig. 7.—SameasFigure5,butfortheI13model. Fig. 9.—SameasFigure5,butfortheAMmodel. Inthiscase,
nojetinflowisshownsincethejetsareneveractivated.
ofjetaction. Really,weareinterestedinenergyflowsre- curacy with which we account for energy changes on the
sulting from the jet-ICM interactions rather than those grid(theexpectationvalueiszero),whilethedottedlines
established by the details of the ICM initialization, so showthechangeineachenergytypeasusual. Nodashed
this background relaxation is subtracted away from the lines are included since no jet inflow was introduced. As
measured change in energy (dotted lines) in each of Fig- expected, the AM model relaxes by gaining kinetic and
ures 5 - 8. To calibrate this background energy flow, we thermal energy at the expense of gravitational and mag-
usedtheresultsoftheAMmodel,whichfeaturednojets. netic energy. The amounts of energy that change form
Figure 9 shows the energy flow evolution for this model. intheAMmodelareonlyapproximately1%ofthetotal
The solid line in each plot in Figure 9 illustrates the ac- added jet energy in the ST model, but this amount is
8
energy. Since little gravitational energy is introduced
at the jet orifice, the increase in gravitational energy is
almost entirely a result of energy exchange in these sys-
tems. Finally, the measured magnetic energy evolution
alsofeaturesaninterestingshape,suddenlyincreasingin
each figure. This is representative of the fact that the
measured magnetic energy change on the grid is actu-
ally negative at early times and so is not shown on these
plots; a point we will discuss shortly.
Before we discuss the details of each model, we should
first introduce another measure of energy flow in these
systems. By incorporating the passive color tracer, we
can determine what fraction of the measured change in
energy on the grid takes place in the ICM. This is ac-
complished by measuring (1−C )∆E, where C =1 for
j j
jet material and C = 0 for the ICM. Figure 10 shows,
j
for the four jet models, this fraction for each different
type of energy. Again, the background change in energy
due to the relaxation of the ICM is removed so that the
includedchangesareultimatelytheresultofjet-ICMin-
teractions. The most significant trend to note is that all
modelsareefficientattransferringenergyfromjetstothe
ICM. As we found in O’Neill et al. (2005), efficient en-
ergytransferseemstobeageneralcharacteristicofthese
systems, and all jet models have transmitted ∼ 60% or
Fig. 10.— Energy deposited by jets into the ICM over time, as
more of their energy to the ICM by the end of the sim-
a fraction of total energy added to the grid. TE: thermal, KE:
kinetic, GE: gravitational, BE: magnetic. The model name is in- ulations. Significantly, thermal energy always represents
dicatedintheupper-rightportionofeachplot. the largest fraction of energy added to the ICM for all
models. Unsurprisingly, given the large β in the jet in-
worth tracking since it exceeds the total added jet mag- flow, the magnetic energy always represents the smallest
netic energy in that model, for example. fraction of energy added. The exact distributions of en-
FocusingnowonadiscussionofFigures5-8,wepoint ergyvaryaccordingtomodel,however,sowewilladdress
out some general features of energy flow in these sim- them individually.
ulations. Unsurprisingly, the inflowing jet energy in all Focusing for the moment on steady jets, represented
cases is dominated by the kinetic energy component (il- bytheSTmodel,thereareseveralinterestingfeaturesto
lustrated by the dashed line being just below the solid note. Similar to what we found in O’Neill et al. (2005),
lineintheupper-leftplotinFigure5,forexample). Most the active jets efficiently convert kinetic to thermal en-
oftheremainderoftheenergyaddedbythejetsisinthe ergy. The measured thermal energy is asymptotically a
form of thermal energy, with relatively minor magnetic factorof∼2largerthanthermalenergyaddedbythejet,
and gravitational components. The gravitational com- whichissimilartotheamountofkineticenergylostfrom
ponent (dashed line in lower-right plot), in particular, thejetinflow. Fromtheupper-leftpanelofFigure10,we
is roughly constant for active jets, representing the in- seethat∼40%ofthetotalenergyaddedtothegridends
troduction of jet material in the gravitational potential upasthermalenergyintheambientmedium,suggesting
uponjetinitialization. Wenotethatthethermalcompo- thatthesesteadyjetsefficientlyheattheirenvironments.
nentoftheinflowingjetpower(dashedlineinupper-right Althoughitisclearthatthisenergyisbeingaddedtothe
plot) asymptotically reaches a somewhat larger fraction ICM, we also seek to discover where in the ICM this en-
of the total jet luminosity for all active jets as time goes ergyappears. AsVernaleo&Reynolds(2006)andHeinz
on. This is caused by the sides of the jet cylinder al- et al. (2006) demonstrated, the location of deposited en-
lowing material back across that boundary, and is thus ergy is at least as significant as the amount of energy
an artifact of the way in which we launch our jets. The depositedforasystemtooffsetcooling. Figure11shows
calculatedjetpowerisreducedbythisamount,however, illustrationsoftheentropydensitys=T/ρ2/3 fortheST
so that the depictions of jet power are an accurate rep- and I13 models. Specifically, we apply the color tracer
resentation of the net inflow from the jet region. and show ∆s = s − s for C < 0.10, which removes
0 j
Therearealsoseveralglobaltrendspresentinthemea- the initial ambient entropy s and focuses only on en-
0
suredchangesonthegrid(dashedlines)forthejet-driven hancementsintheICM.Lookingattheupper-leftpanel,
models. First, we note that in all models the measured most of the visible ∆s forms a sheath around the jet.
thermalenergyexceedsanyoftheothercomponents. Re- This is caused by a mixture of jet-ICM mixing at the
gardless of the details, all models generate substantially boundary (jets have low density, so at a given pressure,
morethermalenergythanisintroducedthroughthejets. high entropy) and ICM entropy increases resulting from
Additionally,wenotethatallmodelsfeatureappreciable shock heating. In the saturated image (upper right), we
increasesinthegravitationalenergy. Unlikeourprevious can see the effects of the bow shock, although the shock
work(O’Neilletal.2005),thesesimulationsincludeare- discontinuity itself is not visible. These images suggest
alistic model of gravity in which the total gravitational that much of the irreversibly added energy measured in
energy on the grid is comparable to the total thermal
9
Fig. 11.—Color-selectedentropydensity(T/ρ2/3)crosssectionsfortheST(upperpanels)andI13(lowerpanels)modelsatcomparable
jetlengths. Entropyisonlyshownformaterialthatisatleast90%ICMbymass,andtheinitialentropyissubtractedoff. Higherintensities
correspondtohigherentropies,althoughtheintensitiesintherighttwopanelsaresaturatedtopermitvisualizationoftheinfluenceofthe
bowshock.
the ICM in the ST model probably is transferred and or entrainment — or through the ICM reacting to the
remains near the jets themselves rather than uniformly added heat and local pressure. In the ST model, drag-
filling the shocked ICM region. ging (i.e., as in a wake) should not play a role since the
Figure5alsoshowsthatthemeasuredgravitationalen- jet is never disconnected from the source. Since the pas-
ergy change represents an increasing fraction over time sive variable appears to have a sharp boundary in this
ofthetotaladdedenergyforsteadyjets,althoughitstill model(whichyoucanseefromthemass-fraction-selected
comprises less than 10% of the total energy by the end entropy in Figure 11), the ICM is also not significantly
of the simulation. Since the jets are light with respect entrained. Any lifted ICM material would have to be
to the ambient medium, they contain a very small frac- located between the jet/ICM contact discontinuity and
tion of the total gravitational energy. In fact, you would thebowshock. Inthedirectionofjetpropagationforan
have to lift jet material 100 times higher than core ICM active jet, this is a very small region and so not likely to
material for the same energy gain. Rather, this mea- contributemuch. SinceweknowthattheICMisheated,
sured change is caused by the jets driving ICM material this is the most likely source of the increase in gravita-
higher into the gravitational potential. Figure 10 also tional energy for this model.
supports this interpretation since the fraction of gravi- The evolution of the magnetic energy in the ST model
tational energy added to the ambient medium (∼ 5%) is interesting since magnetic energy is initially lost from
is comparable to the total fraction estimated from Fig- the system and later increases. In fact, this is a general
ure 5. The increase in gravitational energy can either be characteristic of all four jet models. Although the to-
accomplished mechanically — through lifting, dragging, tal magnetic energy budget is tiny with respect to the
10
Fig. 12.— Length evolution of our model jets. Solid lines show Fig. 13.— A comparison of the magnetic energy evolution of
themaximumextentinthex-directionasmeasuredbythepassive theSTmodeltotheICMmagneticfieldstructure. Thesolidline,
variable. Dotted lines indicate the position of the bow shock as corresponding to the values on the y-axis, shows the change in
a function of time. Values are averaged over the two jets in each magnetic energy on the grid in the ST model. The dotted line
simulation. (scaled for comparison) shows the magnitude of the ICM field in
thedirectionofjetpropagation.
other forms of energy (making the accurate subtraction expansion.
of the ICM more crucial), we still want to understand Interestingly, there is a correlation between changes in
howmagneticfieldsevolve,particularlysincethesefields magnetic energy and jet encounters with the structure
are an essential component of the synchrotron emission of the ICM magnetic field. As Figure 4 shows, the ra-
we observe. There are two possible physical scenarios dial distribution of magnetic field is not uniform, but
onecanimaginetoachieveanetdecreaseinmagneticen- rather peaks in magnitude every 50 kpc or so. Using the
ergy after the ICM relaxation has been subtracted. One measured jet lengths in Figure 12, which we will discuss
is that the stronger central ICM fields get advected up- further in Section 3.2, we can translate the time evolu-
ward to where they can more easily expand, reducing tion of magnetic energy to evolution as a function of jet
their magnetic energy. This seems implausible since the length. Figure 13 shows the relationship between the
ST model does not advect much ambient material with- change in magnetic energy and the ICM magnetic field
out compressing it. Also, the observed effect is almost in the direction of jet propagation. The two quantities
immediate, and so operates at early times. The second appear to be correlated — or anti-correlated — with a
possibility is that the ambient fields reconnect with the slight offset. The causal reasons for this are interesting
jet fields and then relax. Although reconnection as a since it is not obvious that the structure of the globally
processwouldnotlikelyleaddirectlytoasignificantloss weak ICM field should be detectable in the energy flow.
of magnetic energy, it has the potential to create new AnimationsassociatedwithFigure14showthemagnetic
fieldgeometriesthatcouldfacilitatemagneticrelaxation. fieldevolutionfortheSTandI13models. Ineachcase,it
This is the more plausible of the two scenarios since the is clear that the ICM magnetic energy is enhanced each
jet fields certainly reconnect with the surrounding ICM time the jet passes through a strong magnetic pressure
(see the discussion in Section 3.3). Unfortunately, it is gradient in the ICM. These waves of enhancement lag
less obvious that this process would necessarily lead to slightly behind the jet intersection of a high-field region,
a net reduction in magnetic energy since the stretching suggesting that the solid line in Figure 13 is correlated
and twisting of reconnected fields could also amplify the with, but lags slightly behind, the dotted line. Since a
fields(see,forexample,Figure18). Afterthisinitialdip, substantial fraction of the magnetic energy on the grid
however, it is clear that the overall magnetic energy in is located in these regions, it makes sense that their per-
the system is increasing. Some of this is compression turbations would manifest themselves in the measured
of the ICM magnetic field by the jet/cocoon/bow shock, magnetic energy changes.
butFigure14suggeststhatmuchofitissimplyPoynting Weturnnowtoadiscussionofenergeticsinthetwoin-
flux introduced by the jet inflow. The animations asso- termittentjetmodels,I26andI13. AsFigure12andthe
ciated with Figure 14 further suggest that the magnetic animations associated with Figure 1 illustrate, the ap-
energy in the cocoon is replenished by the jet since the pearancesofthetwointermittentjetmodelsdeviatefrom
field in the cocoon is reduced in magnitude during peri- oneanotherveryearlyon. Thisisduemostlytothefact
odsofjetquiescence,presumablyasaresultofadiabatic that the I26 model manages to form a well-defined low-