peng wang and tom abel- magnetohydrodynamic simulations of disk galaxy formation: the magnetization...

8/3/2019 Peng Wang and Tom Abel- Magnetohydrodynamic Simulations of Disk Galaxy Formation: The Magnetization of the

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Draft version December 5, 2007Preprint typeset using LATEX style emulateapj v. 12/14/05

MAGNETOHYDRODYNAMIC SIMULATIONS OF DISK GALAXY FORMATION:THE MAGNETIZATION OF THE COLD AND WARM MEDIUM

Peng Wang and Tom Abel1. Kavli Institute for Particle Astrophysics and Cosmology,

Stanford Linear Accelerator Center and Stanford Physics Department, Menlo Park, CA 940252. Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106

Draft version December 5, 2007

ABSTRACT

Using magnetohydrodynamic (MHD) adaptive mesh refinement simulations, we study the formationand early evolution of disk galaxies with a magnetized interstellar medium. For a 1010 M halo withinitial NFW dark matter and gas profiles, we impose a uniform 109 G magnetic field and follow itscollapse, disk formation and evolution up to 1 Gyr. Comparing to a purely hydrodynamic simulationwith the same initial condition, we find that a protogalactic field of this strength does not significantlyinfluence the global disk properties. At the same time, the initial magnetic fields are quickly amplifiedby the differentially rotating turbulent disk. After the initial rapid amplification lasting 500 Myr,subsequent field amplification appears self-regulated. As a result, highly magnetized material beginto form above and below the disk. Interestingly, the field strengths in the self-regulated regime agreeswell with the observed fields in the Milky Way galaxy both in the warm and the cold HI phase and

do not change appreciably with time. Most of the cold phase shows a dispersion of order ten inthe magnetic field strength. The global azimuthal magnetic fields reverse at different radii and theamplitude declines as a function of radius of the disk. By comparing the estimated star formationrate (SFR) in hydrodynamic and MHD simulations, we find that after the magnetic field strengthsaturates, magnetic forces provide further support in the cold gas and lead to a decline of the SFR.Subject headings: galaxies: formation galaxies: ISM

1. INTRODUCTION

Rapid developments in observational cosmology re-cently established a standard model of cosmology, theCDM model (Spergel et al. 2007). In principle, thismakes the problem of galaxy formation essentially aninitial value problem. Indeed, Gpc-scale N-body simu-

lations of CDM cosmology are revealing the details oflarge-scale structure formation and clustering, matchingobservations in many aspects (Springel et al. 2005). Inthis framework, it should be possible to understand theformation and evolution processes of individual galaxies.For this purpose, it is crucial to understand the structureof interstellar medium (ISM) and the dominant physicscontrolling star formation.

Analytical and semi-analytical studies of disk galaxyformation are useful in understanding the global proper-ties (Mo et al. 1998; Efstathiou 2000; Dutton et al. 2007;Kampakoglou & Silk 2007; Stringer & Benson 2007). Be-cause of the complex nature of galaxy formation and thewide range of physical processes involved, numerical sim-

ulations also prove particularly useful. Hydrodynamicalsimulations of galaxy formation in both cosmological andisolated setups have taught us a lot about the detailedstructure of the galactic ISM and star formation (seee.g. Kravtsov 2003, Li et al. 2005, Springel & Hernquist2005, Okamoto et al. 2005, Tasker & Bryan 2006, Tas-sis et al. 2006, Governato et al. 2007, Wada & Norman2007, Kaufmann et al. 2007, Robertson & Kravtsov 2007for recent studies). However, there are still many physi-cal processes we need to include to make the simulationsrealistic.

Electronic address: pengwang, [email protected]

In this work, we focus on magnetic fields which havenot yet been included in previous galaxy formation sim-ulations. Magnetic fields may significantly influence thestructure and evolution of ISM and has been studiedextensively previously by local magnetohydrodynamic(MHD) simulations of the ISM (Mac Low et al. 2005;

Balsara et al. 2004; de Avillez & Breitschwerdt 2005; Pi-ontek & Ostriker 2007; Hennebelle & Inutsuka 2006)and by observations (e.g. Beck 2007, Crutcher 1999).On a larger scale, the effect of magnetic fields is lessclear because cosmological MHD simulations are still atan early stage. Miniati et al. (2001) performed cosmo-logical hydrodynamic simulations for a passive magneticfield. Cosmological MHD simulations have been done inSPH (e.g. Dolag 1999) in the context of galaxy clusterformation and Eulerian-code simulations are also begin-ning to be explored (e.g. Li et al. 2006). None of thosecalculations have as yet resolved the internal structuresof disk galaxies.

Besides the potential importance in the dynamics of

galaxy formation, galactic magnetic fields are a key in-gredient in many other astrophysical problems. For ex-ample, they play a dominant role in the origin and prop-agation of cosmic rays (Fermi 1949; Strong et al. 2007),which may also be dynamically important in galaxy for-mation (Fermi 1954; Enlin et al. 2007). Furthermore,magnetic fields may be important in the dynamics of su-pernova remnants (Balsara et al. 2001), in regulating theinternal turbulence of giant molecular clouds by driv-ing outflows (Li & Nakamura 2006; Banerjee & Pudritz2007) and in the collapse of individual molecular cloudcores (Mouschovias 1987; Balsara et al. 2001; Tilley &Pudritz 2007; Mellon & Li 2007).

While the importance on the ISM is generally accepted,


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the origin of the galactic magnetic field is still an un-solved problem (see e.g. Kulsrud & Zweibel 2007 fora recent review). The traditional theory of a galac-tic dynamo, the alpha-Omega dynamo, is based on themean field dynamo theory introduced by Steenbeck etal. (1966) and later developed by Parker (1971) andVainshtein & Ruzmaikin (1971). Ferriere (1992) hasextended this galactic dynamo theory to include the ex-

pansion of supernova bubbles. Turbulent amplificationin protogalaxies have also been studied (Pudritz & Silk1989; Kulsrud et al. 1997). Those theories are builton kinematic dynamo. A different paradigm for disk dy-namo comes with the discovery of the magneto-rotationalinstability (MRI) (Chandrasekhar 1961; Balbus & Haw-ley 1991). In MRI, field amplification is not only a con-sequence of the turbulent velocity field, it also producesthe turbulent velocity field itself. It is still unclear what isthe dominant mechanism that amplifies the galactic mag-netic field. In contrast to the case of dynamos in planets,because of the enormous inductance of the ISM (Fermi1949), there is no decaying problem for the galactic field.So the essential part of the problem here is indeed only

the initial amplification.In the following, we first describe the numerical al-

gorithms and initial conditions. Then in section 3 wepresent the results. Sections 4 and 5 are devoted to dis-cussion and conclusions, respectively.

2. NUMERICAL METHODS AND MODELS

2.1. MHD Algorithm

We have developed a 3D adaptive mesh refinement(AMR) MHD code based on the cosmological AMR hy-drodynamics code Enzo (Bryan & Norman 1997; OSheaet al. 2004). We use the Dedner formulation of MHDequations (Dedner et al. 2002). This conservative for-mulation of the MHD equations uses hyperbolic diver-

gence cleaning. It has also been used in several otherrecent AMR MHD codes (Matsumoto 2006; Andersonet al. 2006). The conservative system is discretized bymethod of lines. The Riemann solver and reconstruc-tion schemes have been tested extensively in Wang etal. (2007) for relativistic systems. Since our Rie-mann solvers and reconstruction schemes are designedto work for any conservative systems, when applying itto the Dedner MHD equations, we only redefine the con-servative variables. Using these high-resolution-shock-capturing (HRSC) schemes designed for conservative sys-tems ensures the conservation of mass, momentum andenergy, as well as obtaining correct shock positions andvelocities (LeVeque 2002). Interested readers can find

more detailed descriptions of our numerical algorithmsin Wang et al. (2007) and references therein. In thiswork, reconstruction is done using the piecewise linearmethod (PLM) (Van Leer 1979). Fluxes at cell inter-faces are calculated using the local Lax-Friedrichs Rie-mann solver (LLF, Kurganov & Tadmor 2000), whichis free of the carbuncle artifacts present in some othercontact-capturing Riemann solvers (Quirk 1994). Timeintegration is performed using the total variation dimin-ishing (TVD) second order Runge-Kutta (RK) scheme(Shu & Osher 1988). The code inherits all of Enzosparallel AMR capability as well as self-gravity, chem-istry, cooling and radiative transfer routines. We havetested the code using various standard hydrodynamics

and MHD test problems including the 1D Brio-Wu prob-lem, 2D MHD rotor, 2D Hydro and MHD Rayleigh-Taylor problem, 3D Sedov-Taylor problem, 3D Larson-Penston isothermal collapse problem, etc.

2.2. Cooling

A cooling function is used to calculate the radiativeenergy losses down to temperature 300 K. We use thecooling function fitted by Sarazin & White (1987) forT > 104 K and by Rosen & Bregman (1995) for 300 K 1 1026 g cm3.

3.1. Disk Structures

As can be seen in Fig. 1, the global disk structures arequalitatively similar in both the Hydro and MHD cases.The density projections show filamentary structures andhigh density blobs located along those filaments. Werewe to Include molecular hydrogen chemistry and cooling,these blobs would presumably resemble giant molecularcloud complexes. This is similar to the density distri-

butions of HI gas and GMCs in observations of nearbygalaxies such as M33 (Engargiola et al. 2003), which alsoshow good correspondence between the filaments and thelocations of the GMCs. The temperature projectionsshow that those filaments are all cold, with T 300 K.The lower density disk material has a temperature ofT 104 K. In the MHD run, Fig. 2 shows that much ofthe cold gas is already strongly magnetized.

In a self-gravitating disk, the vortex mode is the fastestunstable mode with growth rates 2 3 times larger thanthe spiral density wave mode (Mamatsashvili & Chage-lishvili 2007). In an isolated non-rotating fluid, theperturbation mode is divided into vortical and divergenttypes according to the Helmholz decomposition. And itis mainly the energy in divergent flows that will be dis-sipated in shocks. In a differentially rotating disk, thevortex mode is a combination of vortical and divergentmotions. Thus the morphology and statistical propertiesof large scale shocks which form the filaments are differ-ent in those two cases. However, there seems to be noquantitative comparative studies on this problem that weare aware of.

To see how self-gravity affects the growth of vortexmodes, we have carried out experiments with only theexternal NFW potential, neglecting the self-gravity ofthe gas. Analytical calculations in the thin sheet ap-proximation predict that the growth rate of vortex modeis orders of magnitude smaller without self-gravity (Ma-


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Fig. 3. Disk profiles for the Hydro run (thin lines) and the MHD run (thick lines) at t = 1.088. Top left gives the mass profile, topright the surface density profile, bottom left the velocity dispersion, bottom right the rotational velocity profile. In the mass profile plot,the thin dashed line is the mass profile of the static NFW halo. In the surface density profile plot, the dashed line is the threshold gassurface density for star formation 5 Mpc2 as observed by Martin & Kennicutt (2001). In the rotational velocity plot, the thin dashed

line is the rotation velocity corresponding to the static NFW halo v,NFW(r) = [GMNFW(r)/r]1/2.

Fig. 4. Toomre Q profiles for Hydro run (thin line) and MHDrun (thick line) at t = 1.088.

matsashvili & Chagelishvili 2007). Indeed in this case wesaw that the disk shows much less filamentary structuresand instead develops tightly wound spiral density wavepatterns. Thus the vortical turbulent motion in the fidu-cial cases is clearly driven by self-gravity. Another inter-esting implication of this experiment is that many localgalaxies do show tightly wound spiral patterns. Henceour experiment suggests that the gravitational force in

the gas disks of those galaxies is dominated by the stars.This implies that for disk galaxies at high redshift thatare still self-gravitating, the vortex modes may be thedominant structures regulating molecular cloud and starformation within them. In the following discussion wewill call the turbulent velocity field seen in our simula-tions gravity-driven turbulence (Gammie 2001; Wada etal. 2002).

As also can be seen in Fig. 1, the MHD run has a some-what thicker disk than the Hydro run. This shows att = 1.088 Gry, magnetic pressure is already large enoughto provide additional support to the vertical force bal-ance, which can also be seen from Fig. 2.

The pressure projections, show that the cold fila-ments have higher pressure than the warm gas and thecores of the filaments have even higher pressure. Theoverpressure in the cores is expected as they are self-gravitating (Larson 1981). The higher pressure in non-selfgravitating cold gas is consistent with previous simu-lations of isolated disk galaxies (Tasker & Bryan 2006).This result is different from the idea that cold gas isformed by isobaric thermal instability in the galactic disk(Field 1965). One important difference with the originalthermal instability analysis is that instead of cosmic rayheating, the heating sources in our simulation are pdVwork, numerical viscous heating and numerical resistivity


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in the MHD run, all controlled by the gravity-driven tur-bulence velocity field in the disk. Thus in our simulationsshow that gravity-driven turbulent heating and radiativecooling will create a non-isobaric two phase medium. Inthe old isobaric theory, there is no prediction for themorphology and fraction of the cold phase. But in theturbulent heating scenario, one would expect the coldgas to have filamentary structures, which are naturally

produced by a supersonic turbulent velocity field.Fig. 3 shows the azimuthally averaged radial profiles

of mass, surface density, velocity dispersion and rota-tional velocity for both simulations at t = 1.088 Gyr.The Hydro and MHD calculations have similar densityprofiles and remarkably similar mass profiles. This is be-cause magnetic field amplification happens after the diskis formed so the initial weak magnetic field will not af-fect significantly the dynamics of disk formation. In bothcases, there are no sharp boundaries at the disk edge insurface density profile. From the mass profile one seesthe disk radius to be 3 kpc for both simulations. Weoverplotted the threshold gas surface density for star for-mation

5 Mpc

2 derived from observations (Martin

& Kennicutt 2001). It is interesting to see that it is just near the disk radius. In the mass profile plot, thedark matter halo mass profile is also shown. It showsthat the dark matter mass dominates baryon at r > 1kpc. This explains why the rotational velocity profile isroughly flat at r > 1 kpc with v 50 km/s, appropriatefor the chosen NFW mass profile. In the rotation velocityplot, there is a big trough at 1 kpc in both the Hydroand MHD curves. This is because of the fact that thereare some big clumps at this radius whose spin providesa large cancellation to the mass-averaged rotation veloc-ity. This also gives the trough in Toomre Q parameterdiscussed below.

The bottom left panel of Fig. 3 shows the velocity dis-

persions calculated by = < v 2 > < v > where< > represents mass-weighted azimuthal average. Thevelocity dispersion has been argued to remain roughlyconstant in self-regulated regions of disks (Silk 1997),but this is still controversial (Ferguson et al. 1998).We can see that across the disk the velocity dispersionshave about an order of magnitude fluctuations centeredaround 510 km/s. So taking it to be constant is only arough approximation. However, Silk (1997) argued thatthe hot phase is the key in creating the self-regulatedstage and determining the global velocity dispersions.Since hot phase is completely missing in our simulations,the situation might be different once supernova feedbackis included.

Fig. 4 shows the Toomre Q parameter Q = /(G)(Toomre 1964), where is the angular velocity, isthe velocity dispersion, is the surface density. In theoriginal Toomres formula, is replaced by the epicyclicfrequency = (rd2/dr + 42)1/2. We used becauseour disk is highly turbulent, different averaging meth-ods for calculating would give different . In practice holds to within a factor of two. As can be seenfrom Fig. 4, throughout the disk Q fluctuates around1, implying that the disk is Toomre unstable. Such aroughly constant Q suggests that the gravity-driven tur-bulence in the disk is in a quasi-stationary state at thistime (Gammie 2001; Wada et al. 2002). This is also

Fig. 5. Mass-weighted joint temperature-B PDF at t = 1.088Gyr. The total mass in the disk is 109 M, as can be seen fromthe mass profile plot of Fig. 3. The binsize is 0.02 in log space.The plus signs are the averaged magnetic field strength in everytemperature bins.

Fig. 6. Mass-weighted joint density-B PDF at t = 1.088 Gyr.The solid straight lines are B2 and B0.8. The color scaleand binsize are the same as that of Fig. 5. The plus signs are theaveraged magnetic field strength in every density bins.

supported by the quasi-stationary density PDF (see sec-tion 3.3). Beyond the disk radius, Q rises sharply to> 1.

3.2. The Growth and Structure of the Magnetic Field


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Fig. 9. Mass-weighted average of the toroidal magnetic fieldstrength at t = 272 Myr (thin solid line), 544 Myr (dotted line),680 Myr (dashed line), 816 Myr (dash-dotted line) and 1.088 Gyr

(solid line)

this implies that in cold gas, fluctuates by two orderof magnitudes. Thus the average gives only limitedinformation for the magnetization of cold gas. Similarresult has also been found in local MHD simulations ofmolecular cloud (Tilley & Pudritz 2007). Furthermore,Fig. 5 also shows that magnetic field strength is similarin warm and cold medium (see also Fig. 2). Then sincethermal pressure is different in those two phases, this re-sults in the vast difference of plasma beta in those twophases.

Fig. 6 shows the mass-weighted joint density-B PDF.It shows that the density-B relation roughly follows

power law relation B rhok

. For the high density diskgas, k = 0.5. This implies that the Alfven speed is sim-ilar in those density ranges. This is also consistent withthe observations ofB relation in dense molecular gas(Crutcher 1999). The scatter in both observational dataand our result are quite large. For the low density halogas, k = 1.25. This results from the rise-up of the lowbeta bubble.

To see how the magnetic field components scale withdensity, Fig. 7 shows the mass-weighted joint density-Band density-Bz plots. It can be seen that both com-ponents have similar density-dependence. In the highdensity gas, azimuthal magnetic component dominatesover the vertical component, similar to what was found

in shearing box simulations of MRI (Hawley et al. 1996).In the low density bubbles, vertical magnetic field dom-inates over the azimuthal component. Interestingly, thisis also consistent with radio polarization observations ofnearby edge-on galaxies (Krause et al. 2006).

To examine the amplification history of the magneticfield, the top panel of Fig. 8 shows the time evolutionof the total kinetic energy, thermal energy and magneticenergy for the disk gas. The magnetic energy increasesexponentially by about five orders of magnitude in thefirst 600 Myr. Afterwards it remains invariant with a sat-urated magnetic field strength in the disk. Some of thenewly amplified field after saturation will rise above thedisk, magnetizing the halo and another fraction of it will

Fig. 10. Density PDF for Hydro (top) and MHD (bottom)runs at t = 272 Myr (thin solid line), 544 Myr (dotted line), 680Myr (dashed line), 816 Myr (dash-dotted line) and 1.088 Gyr (solidline).

be lost by dissipation. Furthermore, after saturation, themagnetic energy is still three orders of magnitude smallerthan the total kinetic energy. This explains in part whyin our simulation magnetic fields did not influence signif-icantly the global disk properties. It could happen thatat higher resolution magnetic fields energy can be am-plified to even larger value due to unresolved small-scale

dynamo effects and may influence the disk more signifi-cantly. Nevertheless, it is encouraging that with the cur-rent resolution, the magnetic field strength in the cold gasalready is very similar to the observed values in the MilkyWay (Crutcher 1999). This is different from the result ofdynamo simulations in MRI-driven turbulence where thetotal magnetic energy in the saturated state is generallylarger than the kinetic energy (Hawley et al. 1996). Theprimary differences are that turbulence in galactic diskis mainly driven by self-gravity. Furthermore, the simu-lated galactic disk has a multi-phase medium. The coldphase controls the saturation. And the warm phase stillhas a subdominant magnetic field in the saturated state.This also contributes to the subdominance of magnetic


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energy.The bottom panel of Fig. 8 shows the time evolution of

the mass-weighted averaged logarithm of plasma beta forall gas, cold gas and warm gas in the disk. For t < 600Myr, the plasma beta in the cold gas quickly deceasesfrom 105 to one and then remains around unity. Themagnetic field in the warm phase is always highly super-critical. This evolution behavior suggests that magnetic

field amplification in disk galaxies is a self-regulated pro-cess. After the plasma beta reaches one in the cold gas,the magnetic fields become dynamically important andbuoyant (Parker 1966). Further growth of magnetic fieldis quenched in the disk. This could be because of insuffi-cient resolution leading to too large dissipation, yet our52 pc resolution run gives very similar result. On theother hand, we can see clearly that magnetic fields beginto rise above the disk after saturation, forming the lowbeta bubbles. This suggests that at least up to 1 Gyr,buoyancy is a relevant mechanism of removing magneticflux from the disk. To further confirm this picture, wehave run simulations that start with a ten times higherinitial magnetic field, i.e. 108 G. They reach the self-

regulating phase earlier and then drive more low betabubbles magnetizing the halo in agreement with the pre-ceding interpretation.

The thick line in Fig. 9 shows the mass-weigthedtoroidal magnetic field at t = 1.088 Gyr. There areregular patterns of field reversal at late times. This isa consequence of field amplification by differentially ro-tating disks (Parker 1979). Fig. 9 also shows the timeevolution of the mass-weigthed toroidal magnetic field.It can be seen that at early times the averaged toroidalfields are small, before they are amplified by disk rota-tion. The amplitude generally decreases with radius be-cause the rotation frequency decreases. A perhaps moresurprising phenomenon occurred between t = 816 Myr

to t = 1.088 Gry, when the toroidal field reversed almostexactly symmetrically.

3.3. Statistical Properties of the ISM

The density PDF is of interest with regards to star for-mation since star formation is thought to be determinedby the high-density tail of the PDF (Elmegreen 2002;Krumholz & McKee 2005; Wada & Norman 2007). Ahigher order (though not necessarily small!) effect is thespatial distribution of mass. When calculating the den-sity PDF, we considered only gas in the disk. Fig. 10shows the time evolution of this density PDF for bothruns. It can be seen that the PDFs for the two cases aresimilar. The MHD calculation has slightly less volume

in dense gas. In both cases the PDF reaches a quasi-stationary state within less than 500 Myr. This quickestablishment of a quasi-stationary PDF suggests thatthe gravity-driven turbulence in the disk reached a quasi-stationary and self-regulated state. This phenomenon, ifrobust even including supernova feedback, may play acrucial role in explaining the the observed universal rela-tion between star formation rate and gas density (Kenni-cutt 1998; Wong & Blitz 2002). Our temperature flooris a feedback on the flow that in reality will be dominatedby HII regions and supernova feedback.

Our density PDFs cannot be fitted by log-normal dis-tributions. At least, they do not show the typical turn-over of a log-normal distribution. This is different from

Fig. 11. Joint Temperature-Density PDF for Hydro (top) andMHD (bottom) runs at t = 1.088 Gyr.The color scale and binsizeare the same as that of Fig. 5. The plus signs are the averagedtemperature in every density bins.

previous simulations of disk galaxies which claimed thatlog-normal is a good fit to the density PDF (Tasker &Bryan 2006, 2007; Wada & Norman 2007). The crucial

difference could be that those studies start from an iso-lated disk while in our case, the disk is built inside-outdynamically. Thus anytime in the evolution of the diskgalaxy (< 1 Gyr) we have simulated, there is low densitygas reaching the disk from outside. It is possible that ifwe continue to evolve the disk long enough that after theinitial halo gas all fell down to the disk it may also obtaina log-normal PDF. However, for a realistic cosmologicalenvironment, infall should never stop completely.

Fig. 11 shows the joint temperature-density PDF.Above > 8 1023 g cm3, the temperature is deter-mined by the floor value which was introduced to ensurethe calculation resolves the Jeans length. So the tem-perature of the gas can at most be trusted below this


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Fig. 12. Time evolution of the volume and mass fraction of cold (T < 103 K) gas for the Hydro (thin line) and MHD (thick line) runs.

density. From this figure, it is also clear that the ISMdevelops a two-phase structure with a warm T 104 Kand a cold T 300 K phase. However, unlike the origi-nally quasi-stationary picture of a two-phase ISM (Field1965), where the phase is characterized by a density andtemperature determined by the balance of heating andcooling, in our case there are three order of magnitudedensity fluctuations in both cold and warm gas. This isdue to the turbulent nature of heating in our simulationas discussed above. However, in both runs, one can stillidentify a good minimum density 1026 g cm3for thecold phase.

Fig. 12 shows the evolution of the volume and massfractions of cold gas in both runs. It can be seen that theamount of cold gas is monotonically increasing in bothvolume and mass fractions. After 600 Myr, both runsreach a quasi-stationary volume fraction of 0.8 andmass fraction 0.99.

3.4. Estimated Star Formation History

Although we did not model star formation explicitly,from the density PDF, we can estimate the star forma-tion rate (SFR) from (Elmegreen 2002; Krumholz &McKee 2005; Wada & Norman 2007)

SFR = fsM(> cr)

tdyn,cr , (2)

where M(> cr) is the total gas mass with density > cr which can be computed from the density PDF,tdyn,cr = 1/(Gcr)1/2 is the dynamical time at cr and fsis a constant specifying star formation efficiency (SFE).

Using cr = 1022 g cm3and fs = 0.01, Fig. 13

shows the star formation history (SFH) of the two simu-lations. It can be seen that while the SFR in the Hydrorun is constantly increasing, in the MHD case,it reachesa peak at t 800 Myr and then decreases. Since t 800Myr is shortly after the time when the average beta inthe cold gas reached unity, this suggests that magnetic

Fig. 13. Star formation history of Hydro (thin line) and MHDrun (thick line) with cr = 1022 g cm3and fs = 0.01.

forces provide further support in the cold gas after theamplification saturated.

Illustrating how sensitive the estimated SFR depends

on cr, Fig. 14 shows M(> cr)/tdyn,cr as a function ofcr. It can be seen that in both cases it does dependon cr. Thus to get the same SFR, the SFE fs shoulddecrease in those densities ranges. However, it is inter-esting that the effect of magnetic field is to make thedependence weaker. From n = 10 cm3 to n = 103

cm3, in the MHD case M(> cr)/tdyn,cr increases byonly a factor of 4.

The current analysis of the SFR may be a poor pre-dictor of what happens when molecular cloud and starformation and feedback are included more realistically.However, the analysis in this section does suggest thatafter the magnetic growth saturates in the disk, magneticfields can significantly influence GMC formation which


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Fig. 14. Ratio of M( > cr)/tdyn,cr as a function of crcomputed from the density PDF for Hydro (thin line) and MHD(solid line) runs at t = 1.088 Gyr.

occurs in the strongly magnetized cold and dense gas.

4. DISCUSSION

Using the Spitzer values for viscosity and resistiv-ity, the magnetic Prandtl number of the warm ISM is 1011. Thus, small-scale dynamos are expected to de-velop. However, those small-scale dynamo effects are notcaptured by our resolution. Nevertheless, previous sim-ulations (Schekochihin et al. 2004) have shown that forvery high magnetic Prandtl number MHD turbulence,the growth of the small-scale magnetic fields is controlledby the rate of strain at the viscous scale and resulted inmagnetic energy tending to pile up at the much smaller

resistive scale. This is because for small scale dynamos,the magnetic field seems to retain its folded structure insaturation with direction reversals at the resistive scale(Schekochihin et al. 2002). So small-scale dynamo pro-cesses may not be important for the large-scale growthof galactic magnetic field. At least, the situation is un-certain. Even if small-scale dynamos would contributesome fraction to the large-scale growth of the galacticmagnetic field, our simulations of magnetic field growthwould be a lower limit. Since we find that the field canbe amplified to the observed value quickly, for the phe-nomenology of galaxy formation, it seems sufficient notresolve small-scale dynamos at the current level of de-tails.

de Avillez & Breitschwerdt (2005) have discussed theresults of 3D AMR MHD simulations of a local patch ofISM with sufficient vertical height to include the effect ofhalo-gas circulation. They find that gas transport to thehalo is not significantly influenced by the magnetic fieldand they do not find the low beta bubble as seen in oursimulations. We think the crucial difference of our sim-ulation from theirs is the presence of global differentialrotation, which dominates magnetic field amplificationin our simulations.

An important limitation of the presented calculationsis that we neglected supernova feedback while their dy-namics could be important for amplifying the magneticfield in the ISM (Ferriere 1992). More importantly, su-

pernova feedback will create the hot phase of the ISM(McKee & Ostriker 1977), which may play a crucial rolein regulating star formation (Silk 1997). Future workneeds to include this supernova feedback to study the im-portance of superbubbles on the structure of ISM, globalstar formation and magnetic field amplification. Anotherlimitation is that we neglected cosmic ray pressure, whichmay be dynamically important. For example, cosmic ray

pressure arising from the Alfven wave instability (Kul-srud & Pearce 1969) may prevent the downsliding ofmatter along the magnetic field line (Rafikov & Kulsrud2000).

5. CONCLUSIONS

The simulations discussed in this work have shown thatthe amplification of magnetic fields and its large-scaletopology is a natural part of the galaxy formation pro-cess. Our main findings can be summarized as:

1. In the initial stage of the build-up of a galactic disk,the magnetic field is very weak and does not significantlyaffect the dynamics of disk formation.

2. In a self-gravitating disk, vortex modes driven by

the self-gravity of the gas grow rapidly and trigger thefirst generation of GMCs formation.

3. In the cold and warm phase created by turbulentheating, the pressure is not isobaric. Temperature anddensity fluctuates by about three orders of magnitude inboth phases.

4. The dynamical formation of a galactic disk resultsin a global density PDF that cannot be fitted by a log-normal function.

5. Differentially rotating galactic disks amplify a seedfield of 109 G to microgauss levels in 500 Myr.

6. The growth of galactic magnetic fields is a self-regulated process. The saturation is reached first in thecold phase. After saturation, the field strength agrees

with the observations of Milky Way magnetic fields. Thecold phase, the magnetic field strength fluctuates byabout an order of magnitude which gives larger than twoorder of magnitude fluctuations in the associated plasmabeta. The saturated total magnetic field energy is threeorders of magnitude smaller than the total kinetic energyin the case presented here.

7. The saturation results in highly magnetized mate-rial around galactic disks at all but the earliest cosmicepochs. The halo magnetic fields may significantly in-fluence the halo-gas interactions, accretion of gas in diskgalaxies, galaxy mergers and the interactions of galaxieswith their environment.

8. After saturation, the toroidal field in the disk dom-

inates over vertical components while in the magnetizedhalo, vertical components dominate over toroidal com-ponents.

9. Large scale magnetic field and velocity field arealigned at many places of the disk. This implies thatcloud formation is likely channeled by flow along fieldlines.

10. After saturation, the magnetic field strength issimilar in the cold and warm medium.

11. Global toroidal field reversals develops naturallyin a differentially rotating disk.

12. Magnetic fields can suppress star formation by pro-viding additional pressure support in the cold gas aftersaturation.


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We have presented the first global numerical experi-ments studying magnetic field amplification and evolu-tion during the formation of disk galaxies. We find itencouraging that many of the aspects discussed here areconsistent with evidence from observations (e.g. com-pare with Beck 2007). Being able to include more ofthe relevant physics such as molecular cooling, radia-tion and supernova feedback a realistic treatment of star

formation, and a consistent treatment of cosmic ray ac-celeration transport, will render the next generation ofnumerical models useful for a more comprehensive un-derstanding of the physics of galaxies than the currentstate of the art.

We thank Zhi-Yun Li and Ralph Pudritz for very use-ful comments and suggestions on the draft. We alsobenefitted from useful discussions with Chris McKee andAxel Brandenburg. We are grateful for the hospitality ofKITP and our interactions with the participants of theprogram Star Formation through Cosmic Time whenthis work was carried out. We were partially supportedby NSF CAREER award AST-0239709 and NSF Grant

No. PHY05-51164. P. W. acknowledges support by anOffice of Technology Licensing Stanford Graduate Fel-lowship and a KITP Graduate Fellowship.

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