Large-eddy simulations of isolated disc galaxies with ... · our isolated disc galaxy (IDG)...

Mon. Not. R. Astron. Soc. 000, 121 (????) Printed 5 September 2018 (MN LATEX style file v2.2)

Large-eddy simulations of isolated disc galaxies withthermal and turbulent feedback

H. Braun1?, W. Schmidt1, J. C. Niemeyer1, and A. S. Almgren21Institut fur Astrophysik, Universitat Gottingen, Friedrich-Hund Platz 1, D-37077 Gottingen, Germany2Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

Accepted 2014 June 3. Received 2014 June 3; in original form 2013 December 18

ABSTRACTWe present a subgrid-scale model for the Multi-phase Interstellar medium, Star forma-tion, and Turbulence (MIST) and explore its behaviour in high-resolution large-eddysimulations of isolated disc galaxies. MIST follows the evolution of a clumpy cold anda diffuse warm component of the gas within a volume element which exchange massand energy via various cooling, heating and mixing processes. The star formation rateis dynamically computed from the state of the gas in the cold phase. An importantfeature of MIST is the treatment of unresolved turbulence in the two phases and itsinteraction with star formation and feedback by supernovae. This makes MIST a par-ticularly suitable model for the interstellar medium in galaxy simulations. We carriedout a suite of simulations varying fundamental parameters of our feedback implemen-tation. Several observational properties of galactic star formation are reproduced inour simulations, such as an average star formation efficiency 1 per cent, a typicalvelocity dispersion around 10 km s1 in star-forming regions, and an almost linearrelationship between the column densities of star formation and dense molecular gas.

Key words: methods: numerical - galaxies: ISM - stars: formation - turbulence

1 INTRODUCTION

Stars are a product of a complex sequence of competingand interacting processes on a vast range of spatial andtemporal scales that concentrate initially dilute gas intocompact cores. It is not yet fully understood how theinterplay of all of the processes involved, such as gravita-tional collapse, cooling, turbulence, magnetism, and stellarfeedback, leads to the observed properties of the interstellarmedium and stars in galaxies. A recent review on theproperties of star formation was presented by Kennicutt& Evans (2012). Of particular interest is the mechanismregulating the observed low efficiency of star formation.Measures of the star formation efficiency are gas depletion- or consumption - time-scales dep = Mgas/M, whichrelate star formation to the available gas supply. Wong &Blitz (2002); Evans (2008); Bigiel et al. (2008); Blanc et al.(2009) infer dep 1 2 Gyr in local disc galaxies fromCO- H-, and UV-measurements with resolutions downto 200 pc. Comparable measurements of gas-rich galaxiesby Daddi et al. (2010); Tacconi et al. (2013); Saintongeet al. (2013) and others indicate a significantly shorter timedep 0.5 Gyr, corresponding to a relative gas consumption

? E-mail: [email protected] E-mail: [email protected]

rate per free fall time ff 0.01. According to the KS rela-tion (Schmidt 1959; Kennicutt 1998, and others), the starformation rate is well correlated with the local gas supply.The power-law slope of measured KS relations depends,however, on the tracers used, the resolution achieved, andother observational limitations (e.g. Onodera et al. 2010;Lada et al. 2010; Leroy et al. 2013). Recent observationsshow a good linear correlation between star formation rateand dense/molecular gas (Gao & Solomon 2004; Lada et al.2010; Bigiel et al. 2011). Evans et al. (2009); Murray (2011)showed that the local efficiency ff,MC ' 0.1 in individualactively star-forming molecular clouds is much greaterthan the efficiency on galactic scales, and their depletiontime-scale is considerably shorter (dep < 100 Myr). Thisimplies that molecular clouds convert a sizable fraction0.1-0.4 of their mass into stars during their lifetime (a few10 Myr, see e.g. Blitz et al. 2007; McKee & Ostriker 2007;Miura et al. 2012) before they are destroyed by supernovaexplosions (SNe) and stellar winds.

Actively star-forming molecular clouds are known tobe strongly supersonically turbulent with typical velocitydispersions around 10 km s1 (e.g. Leroy et al. 2008; Stilpet al. 2013), or larger in interacting galaxies (e.g. Herreraet al. 2011). Regulation by supersonic turbulence is a goodcandidate to theoretically explain the observed properties

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of star formation, as it globally supports a molecularcloud against gravity, but locally produces over-densefilaments and knots that may collapse into stars. A varietyof approaches have been developed in the past years toderive star formation efficiencies from statistical propertiesof gravo-turbulent fragmentation inside molecular clouds(Padoan & Nordlund 2011; Krumholz & McKee 2005;Hennebelle & Chabrier 2011; Padoan et al. 2012; Federrath& Klessen 2013, hereafter FK13). As supersonic turbulencedecays on relatively short time-scales of the order of thesound crossing time, it has to be maintained by someproduction mechanism over the lifetime of a molecularcloud. Processes such as large scale shear and instabilitiesin galactic discs (e.g. Gomez & Cox 2002; Wada et al. 2002;Kim et al. 2003; Kim & Ostriker 2007; Agertz et al. 2009;Krumholz & Burkert 2010), accretion of gas on to a galaxy(e.g. Hopkins et al. 2013; Genel et al. 2012; Elmegreen &Burkert 2010; Klessen & Hennebelle 2010), and mergerevents or other galactic interactions (e.g. Bournaud et al.2011; Teyssier et al. 2010) come into question here, butalso local processes like stellar winds (e.g. Wolf-Chase et al.2000; Vink et al. 2000; Vink 2011), radiation pressure(Krumholz & Thompson 2012) and SNe (e.g. Ostriker &Shetty 2011; Agertz et al. 2009; Vollmer & Beckert 2003),or the effects of thermal instabilities (e.g. Wada & Norman2001; Kritsuk & Norman 2002; Iwasaki & Inutsuka 2014)are possible turbulence production mechanisms.

In order to numerically simulate a realistic galaxy asa whole, star formation and the entailing stellar feedbackhave to be taken into account. However, the resolution toproperly follow the evolution inside star-forming clouds ina galactic scale simulation is far from being feasible withcontemporary computational resources. Recent simulationsof isolated disc galaxies (IDG) feature resolutions down to afew parsec or even a fraction of a parsec (e.g. Hopkins et al.2013; Renaud et al. 2013; Dobbs & Pringle 2013; Benincasaet al. 2013; Booth et al. 2013; Monaco et al. 2012), whilesimulations of galaxies from cosmological initial conditionsreach resolutions of some 10 parsec (e.g. Agertz et al.2009; Munshi et al. 2013; Kraljic et al. 2012). To tacklesub-resolution processes, an appropriate subgrid-scale(SGS) model has to be applied. In the last decade a widerange of different models have been devised to effectivelydescribe star formation and stellar feedback (e.g. Agertzet al. 2013; Stinson et al. 2006, 2013; Wise et al. 2012) usingresolved quantities and assumptions about the small-scaleproperties of the ISM.

In galaxy simulations, the star formation rate s is usu-ally modelled using a constant efficiency ff per free fall time,which locally enforces a KS relation

s = ff

ff 1.5, (1)

where is the local gas density and ff 0.5 the localfree fall time. To avoid spurious star formation, additionalconstraints are applied, for example, a threshold for theminimal density required for star formation and a maxi-mal temperature. More sophisticated models distinguishbetween different gas components. Gnedin et al. (2009)suggest to relate the star formation rate to the density of

molecular gas instead of the total gas density. Murante et al.(2010) use a simple multi-phase approach to determinethe fraction of the gas density that is available for starformation. For the simulations presented in this article,we use a multi-phase model for the ISM, including anestimation of the amount of shielded molecular gas (Braun& Schmidt 2012, hereafter BS12). The star formationefficiency in the molecular gas is dynamically computedfrom the numerically unresolved turbulence energy, whichis determined by a SGS model for compressible turbulence(see Schmidt & Federrath 2011, hereafter SF11). Since weincorporate the coupling between resolved and unresolvedscales as turbulent stresses in the Euler equations, ourgalaxy simulations are large-eddy simulations (LES).Moreover, we apply the energy-conserving AMR techniquespresented in Schmidt et al. (2014). The diagonal part ofthe turbulent stresses acts as non-thermal pressure thatusually dominates over the thermal pressure in cold anddense environments. This allows us to apply both thermaland turbulent feedback by channeling a fraction of theSN energy into the production of SGS turbulence energy.As we will show, this has important consequences forthe regulation of star formation. In a way, this is similarto kinetic feedback (see, e.g., Agertz et al. 2013), withthe important difference that we assume that turbulentmotions are mainly excited on length scales below the gridresolution. For the thermal feedback, a small portion of theSNe energy is stored in a non-cooling budget - decayingon a time-scale of 1 Myr - to mimic the effect of hot SNebubbles on sub-resolution scales, while the rest of the gas isallowed to cool radiatively. Although we include only effectsof ionizing radiation from massive stars and SNe II, weare able to reproduce several observational features of starformation and turbulence in quiescent, gas-rich (or highredshift z 2) disc galaxies and star-forming regions.

This paper is structured as follows. First we describethe numerical methods, the SGS model, and the setup ofour isolated disc galaxy (IDG) simulations in Sections 2 and3. In Section 4, we present results from a suite of four sim-ulations with different treatments of feedback, followed byour conclusions in Section 5

2 NUMERICAL IMPLEMENTATION

We carried out IDG simulations using the cosmologicalhydrodynamics code Nyx (Almgren et al. 2013). Nyx,built on the BoxLib software framework, uses AdaptiveMesh Refinement (AMR) to provide higher numericalresolution in sub-volumes of particular interest. Nyx solvesthe standard Euler equations using an unsplit PiecewiseParabolic Method (PPM); additional source terms aretreated via a predictor/corrector scheme. Nyx is capableof following the evolution of different collisionless massiveparticles in the N -Body formalism using a KickDriftKickalgorithm, and (self-)gravity is taken into account using aParticle Mesh scheme with multigrid solver. We extendedNyx to run adaptively refined LES of an IDG as describedin the following.

To handle sub-resolution processes, such as star forma-

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LES of isolated disc galaxies 3

tion, stellar feedback, cooling, and thermal instability, weuse a model based on BS12 with a few minor modifications,MIST (Multi-phase Interstellar medium, Star formation andTurbulence model). The key-features of MIST are the fol-lowing.

Atomic, metal line, and dust cooling, photoelectricheating on dust. Separation of gas into two phases due to thermal insta-

bilities that exchange energy and material via different mix-ing, heating, or cooling processes. The two phases representa diffuse warm component and a clumpy, cold componentof the gas. Balance of effective (i.e. thermal plus turbulent)pressure between the phases is assumed to obtain their re-spective densities. Formation of stars from the molecular fraction of the

cold phase that is shielded from dissociating radiation. Thestar formation rate is computed dynamically from the ther-mal and turbulent state of the gas. Depending on the age of a stellar population the stellar

feedback is applied. Lyman-continuum radiation and SNe IIare taken into account. The SNe not only enrich the gas withmetals, but also deposit kinetic energy and thermal energyinto the gas. The SNe ejecta are treated as an additionalsub-phase of the warm phase that does not cool efficientlyand is gradually mixed with the rest of the warm phase. Unresolved turbulence is coupled to almost all processes

implemented in MIST. Besides the source terms that arerelated to the scale separation for LES, small-scale pressuregradients caused by phase separation and SNe are taken intoaccount.

An overview of important variables and coefficients is givenin Table 1.

2.1 Gas dynamics

As an extension to the standard compressible Euler equa-tions we introduce a new degree of freedom in the form ofthe SGS turbulence energy density K (see SF11) and itssource terms in order to model the behaviour of unresolvedturbulent fluctuations. Furthermore we include the sourceterms as needed for the MIST model. The set of conserva-tion equations for the evolution of gas becomes

t+ (u) = s,SF + s,FB, (2)

(u)

t+

[uu +

(p+

2

3K

)

]= g us,SF + us,FBs,FB,

(3)

(E)

t+

[uE +

(p+

2

3K

)u u

]= u g TI SGS (E e+ ec)s,SF

+

(eSN +

u (2us,FB u)2

)s,FB SN,

(4)

(K)

t+ (uK SGSK)

= SGS SGS + TI Ks,SF+SN +

exs,FB.

(5)

Here is the gas density, u the velocity vector, E the totalspecific energy of the gas, K the specific turbulent SGS-energy, e the specific internal energy, p = ( 1)e thethermal gas pressure (where = 5/3 is the polytropic equa-tion of state parameter), and g the gravitational accelera-tion vector (see Section 2.1.1). The SGS turbulence modelrelated quantities SGS, SGS, and SGS are defined in Sec-tion 2.1.2. For a definition of the star formation rate s,SFsee Section 2.2.2. The stellar feedback rate s,FB, the mass-weighted average velocity us,FB of SN-ejecta, and the spe-cific supernova energy deposit eSN = 6.5 1049 erg/M areexplained in Section 2.3.2. We apply a multiphase model todetermine the specific thermal energy ec(Tc = 50 K) of thecold phase, and the net cooling rate (see Section 2.2.1) andthe turbulence energy production via phase separation TIand via SN feedback energy deposit SN (see Section 2.2.2).To conserve the kinetic energy of SN-ejecta, exs,FB is addedto K (see equation 34 in Section 2.3.2).

2.1.1 Gravity

The massive components in our IDG simulations are darkmatter, baryonic gas, and stars, where the dark matter com-ponent is assumed to be a static halo, and the other two aredynamically evolved. The gravitational acceleration vectorg is computed as the sum of a static acceleration due to thedark matter halo and the negative gradient of the gravita-tional potential due to the dynamical components:

g = gdm dyn, (6)

with the static acceleration gdm (see equation 37 in Sec-tion 3.1). dyn represents the solution of Poissons equation

4 dyn = 4G(dyn dyn), (7)

where dyn = +s, dyn is the mean of dyn, s is the totalstellar density, and G is the gravitational constant.

2.1.2 Hydrodynamical turbulence model

The interaction between resolved and unresolved turbulentvelocity fluctuations is modelled using the SGS turbulencestress tensor , which can be seen as an analogue to theviscous dissipation tensor in the Navier-Stokes equations.With ui,k := ui/xk its components following SF11 read

ij = 2C1(2K)1/2Sij 2C2K

ui,kuj,kul,mul,m

23

(1 C2)Kij ,(8)

where

Sij = Sij 1

3ijd =

1

2(ui,j + uj,i)

1

3ijuk,k (9)

is the trace-free rate of strain, and the grid scale. Thetrace free stress tensor , used in equations (3) and (4),is given by ij = ij 2ijK/3. The SGS turbulence energyproduction rate SGS, the SGS turbulence dissipation rateSGS (which does not appear in equation 4 as it is absorbedinto ), and the SGS turbulence diffusivity SGS are given

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Table 1. Important variables and coefficients.

Symbol Value Description Reference

Hydrodynamics

Total gas density Equation (2)

u Vector of linear momentum density of gas Equation (3)

E Thermal plus (resolved) kinetic energy density of gas Equation (4)K SGS turbulence energy density Equation (5)

c Fractional density of cold phase gas BS12, equation (14)

h Fractional density of hot SNe ejecta Equation (35)H Hydrogen density Equation (13)

He Helium density Equation (13)

Z Metal density Equation (13) 5/3 Polytropic index of equation of state

SGS turbulence

SGS Production rate of SGS energy by turbulent stresses SF11, equation (10)

Turbulent stress tensor SF11, equation (8)C1 0.02 Linear closure coefficient of turb. stresses SF11

C2 0.75 Non-linear closure coefficient of turb. stresses SF11

C 1.58 Dissipation coefficient of SGS turb. energy SF11C 0.65 Diffusion coefficient of SGS turb. energy SF11

MIST

c,pa Average density of cold phase gas BS12

w,pa Average density of warm phase gas BS12ew Specific thermal energy of warm gas BS12

`c Length scale of cold phase clumps BS12s,SF Star formation density BS12, equation 17

s,FB SNe feedback density BS12, Sec. 2.3.2

PN Star formation free-fall time efficiency in cold gas PN11, FK13, equation (19)SN Production rate of SGS energy via SNe feedback Equation 22

TI Production rate of SGS energy via phase separation Equation 21

ASN SNe evaporation coefficient BS12, equation (15)fTI 0 or 1 Thermal instability switch BS12

TI 0.025 Efficiency of SGS energy production by phase separation BS12

SN 0.085 Efficiency of SGS energy production by SNe feedback BS12

eSN 6 1049 erg M1 Energy release per M of SNe II BS12h 1 Myr Decay parameter of the hot SNe gas Sec. 2.3.2

eh 0.1 eSN Specific thermal energy of gas in SNe bubbles Sec. 2.3.2ec e(Tc 50 K) Specific thermal energy of cold gas BS12floss 0.4 Fraction of prestellar mass loss BS12

b 1/3 . . . 1 Compressive factor, density PDF broadening parameter BS12, equation (23) 1/3 Turbulent velocity scaling coefficient of warm gas BS12 0.1 Metal loading fraction of SNe ejecta BS12

Abbreviated references: BS12 - Braun & Schmidt (2012), FK13 - Federrath & Klessen (2013),PN11 - Padoan & Nordlund (2011), SF11 -Schmidt & Federrath (2011).

by

SGS = ijSij , (10)

(SGS) =CK

3/2

, (11)

SGS = C(2K)1/2. (12)

We use the closure coefficients C1 = 0.02, C2 = 0.75, C =1.58, and C = 0.65 as determined by SF11 for compressibleturbulence.

2.2 Non-adiabatic physics

In the following we describe how unresolved physics such asheating, cooling, star formation and stellar feedback was im-

plemented in the code Nyx. For a more detailed descriptionof the underlying model we refer to BS12. As input for thecomputation of the non-adiabatic physics sources we needthe hydrodynamical state, the source terms belonging onlyto the SGS turbulence model and the stellar feedback terms.Contrary to BS12, stellar feedback is considered an exter-nal source in the calculation, as it depends on the stellarpopulation represented by N -body particles but not on thehydrodynamical state. Given the SGS- and stellar feedbacksource terms, the actual sources are calculated by subcy-cling the BS12 model ODEs locally in a grid cell, to resolveall time-scales of relevant processes, particularly the coolingtime-scale, and then averaging the rate of change over thehydro-step. To follow the metal enrichment, we calculatethree hydrogen density H, helium density He and metal

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density Z. Their conservation equations are of the form

Xt

+ (Xu) = Xs,SF

s,Xt

FB

, (13)

where X indicates one of the species H, He, or Z, ands,X/t|FB =: Xs,FB is the ejection rate of that speciesby SNe (see Section 2.3.2).

2.2.1 Cold and warm gas phases

To keep track of the multiphase state in a grid cell, we in-troduce an additional passively advected quantity, the cold-phase fractional density c, from which we can easily recon-struct the warm-phase density w = c. The warm phasethermal energy is given by wew = ecec with a constantspecific thermal energy ec of the cold phase, correspondingto a temperature Tc = 50 K. The conservation equation ofc reads

ct

+ (cu) =c + wfTIew ec s,SF ASNs,FB,

(14)

where

c = cSGS PAHc c

Lyc and

w = radw wSGS PAHw

w

Lyc

are the net cooling rates of the cool and warm phase, respec-tively. c is effectively a heating rate. Material is removedfrom the cold phase and transferred to the warm phase in-stead of increasing uc const.

ASN = 13826

(w,pa

mHcm3

) 45(`cpc

) 65(

cc,pa

) 35

(15)

is the SN cold-phase evaporation coefficient, and TI = 0.025is the efficiency parameter for turbulence production by thethermal instability, if the indicator fTI = 1 (see BS12). HerePAH is the photoelectric heating rate, Lyc the heating ratedue to Lyman continuum radiation from young, massivestars (see Section 2.3.2), and

radw =ww,pa

radw (w,pa, Z, Tw) (16)

the radiative cooling rate, which is interpolated from a cool-ing table. These tabled cooling rates were computed us-ing the photo-ionization program package Cloudy (Ferlandet al. 1998, version 08.00). w,pa and c,pa are the averagedensities of the warm and the cold phase, respectively. Thoseare computed from the fractional phase densities (c andw) and the thermal and turbulent energies (ec, ew, andK) by assuming balance of effective (thermal plus turbu-lent) pressure between the phases at cold clump scale `c,as explained in detail in BS12. Z is the metallicity of thegas. Tw = Tw(ew, Z) and w are the temperature and thefractional density of the warm-phase gas, that is allowed tocool radiatively. Note that Tw and w may differ from Twand w in areas affected by recent SNe feedback. The treat-ment of the third gaseous phase, the hot SNe ejecta, whichis prevented from cooling, is described in Section 2.3.2. Thetotal net cooling rate, used in equation (4), is then given by = c + w.

2.2.2 Star formation rate

Stars are assumed to form from the molecular fraction ofthe gas in the cold phase, fH2c at a rate (Krumholz et al.2009)

s,SF =fH2cPNtc,ff

, (17)

where PN is the formation rate of gravitationally boundcores per free fall time

tc,ff =

3

32Gc,pa. (18)

To calculate PN, we use the Padoan & Nordlund (2011,hereafter PN11) model in the single free-fall formulation ofFK13

PN =(1 floss)r

12crit

2

(1 + erf

[2c 2 log (rcrit)

(82c )12

]). (19)

Here floss = 0.4 is the fraction of mass in gravitationallybound cores lost during prestellar collapse through windsetc., and c =

log (1 + b2M2c) the standard deviation of

the assumed density probability density function (PDF) oflog-normal shape. The broadening parameter b is set de-pending on which of the three production terms of turbu-lence energy in equation (5) is locally the dominant one

max = max [SN,TI,SGS] , (20)

where

TI = wfTITI (21)

is the contribution due to thermal instability, and

SN = s,FBeSNSN (22)

describes turbulence production by SNe. SN = 0.085 is thefraction of the energy released by SN that is deposited inthe form of turbulent energy. We define b by

b =

1/3 if SGS = max

2/3 if TI = max

1 if SN = max.

(23)

Here we assume the large-scale driving SGS to be mostlycaused by shear, the SNe driving to be mostly compressive,and the thermal instability driving to be of intermediatetype. The corresponding values follow from Federrath et al.(2010).

To obtain the turbulent Mach-number Mc of the coldphase, the SGS energy K has to be rescaled from the gridscale to the cold clump scale `c, assuming a Kolmogorovvelocity scaling exponent = 1/3: 1

M2c =2K(`c

)2( 1)ec

. (24)

1 We assume `c to be the largest scale represented in the cold

phase. Consequently, the scaling of the turbulent velocities is ap-

plied to those scales, on which only the warm phase exists. Tur-bulence in the warm gas is usually subsonic or transonic at most.

In this regime the assumption of a Kolmogorov-type scaling be-

haviour with coefficient = 1/3 seems valid.

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6 H. Braun et al.

The critical over-density ratio rcrit = crit/c,pa, abovewhich a bound object is formed, is given by (FK13)

rcrit = 0.00675 2K

(`c

)2Gc,pa`2c

M2c . (25)

The molecular fraction of cold gas fH2 is computed fromthe cold and warm phase fractional densities and energies,assuming effective pressure equilibrium, using a Stomgren-like approach. The penetration depth of impinging radiationinto a spherical cold clump of diameter `c is determined bythe balance between H2 production and dissociation due toUV-photons. The dissociating radiation field I is assumedto be homogenous and isotropic. However, in dense, cold en-vironments, which are identified by c,pa > 10 mH cm3and T (e, Z) < 1000 K, I is dimmed by a factor

fI =1

3+

2

3max

[1 +

TI T (e, Z)TI

, 0

], (26)

with TI = 500 K, because of the assumed shielding fromradiation by the environment.

The assumption of a log-normal shaped density PDF,which is an essential part of the theory of PN11, appliesto turbulence in isothermal gas. For a consistent definitionof the internal energy of the cold phase in MIST, an adi-abatic exponent = 5/3 is required. However, we assumea constant average temperature of the cold phase becauseof processes which are not explicitly treated. Both observa-tional and numerical studies on the density PDFs show thatthe density PDF of the cold phase of the ISM is well ap-proximated by a log-normal PDF (e.g. Hughes et al. 2013;Schneider et al. 2014). Although a power-law tail is gener-ally found for actively star-forming clouds in which densecores undergo gravitational collapse, FK13 point out thatthis does not significantly affect the star formation efficiencybecause the log-normal turbulent density fluctuations feedthe collapsing gas that populates the power-law tail at highdensities. Despite of the underlying inconsistency, all cur-rently available analytic models for the calculation of starformation efficiencies, including PN11, are based on this as-sumption. Substituting PDFs with power-law tails into thesemodels does not amend the problem because this would leadto divergent integrals. As a consequence, the construction ofconsistent models of the star formation efficiency is an openproblem.

2.3 Implementation of stellar particles

2.3.1 Stellar particle creation

A particle is characterized by its position xp, mass mp, ve-locity up, and an arbitrary number of additional properties.To handle the dynamical evolution of stars, we implementeda particle type with three additional properties: the initialmass mpi, creation time tpc, and metallicity Zp, which areneeded for the application of stellar feedback. A stellar par-ticle does not represent a single star, but a single stellarpopulation with a normalized initial mass function (IMF)dN/dm (where N is the number of stars of individualinitial mass m per solar mass of stellar population) scaledby mpi. We use the IMF of Chabrier (2001).To avoid the repeated creation of particles in all cells where

dots,SF > 0, we introduce a stellar density field s,m, thatacts as an intermediate buffer for the stellar mass. This istreated as an passively advected quantity with respect tothe hydro-solver, but massive with respect to gravity. Itsconservation equation reads

s,mt

+ (s,mu) =s,SF s,mt

SP

, (27)

where s,m/t|SP represents the mass transfer from s,minto stellar particles. The total stellar mass density is givenby s = s,p + s,m.A pair of stellar particles p1 and p2 is created in thecell centre, if the agglomerated s,m exceeds the thresholds,m,max 2 (corresponding to a minimum particle pairmass 2mp,min = s,m,max

3 40 M for a cell size of 30 pc), the mass is removed from s,m. The propertiesof the new particles are

m1,2p = s,m3/2,

u1,2p = u urnd,

m1,2pi = s,m3/2,

t1,2pc = t+ dt/2,

Z1,2p = Z/.

(28)

Here dt is the hydro time-step, and urnd is a random veloc-ity (in opposite directions for p1 and p2 to conserve totalmomentum). This random velocity component is intendedto reflect the unresolved motions of the cold clumps, whichthe stars originate from. Its absolute value is drawn from aGaussian distribution with expectation value 0 and variance2urnd

2urnd = furnd2K

(1

(`c

)2). (29)

The fudge factor furnd = (mpmp,th)/m2p is designed

to obtain f2urndurnd as random velocity component at theend of the particle growth phase, when the final mpi (mpmp,th)1/2 is reached. mp is the mean mass of allstellar particles and mp,th the upper threshold mass for par-ticle growth.A newly created stellar particle p collects the stellar massin s,m along its path [using a Nearest Grid Point algorithm(NGP)]. Its properties are updated using

mp = mp + s,m3,

up =upmp + us,m

3

mp,

mpi = mpi + s,m3,

tpc =tpcmp + (t+ dt/2)s,m

3

mp,

Zp =Zpmp +

Zs,m

3

mp,

xp =xpmp + xs,m

3

mp.

(30)

The final mass is reached, when either mp > mp,th ' 20 mp,min or (t tpc) > 2Myr.

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2.3.2 Feedback mechanism and hot phase

We consider the two physical stellar feedback processes, Ly-man continuum heating and SNe explosions, and use theequations in BS12 to compute their contributions to thesource terms for the update of the hydrodynamical state.The stellar mass sink field s,m acts on the gas only via Ly-man continuum heating Lyc,m, where we assume zero ageof the stellar population it represents. The amount of feed-back (the mass of SNe ejecta mp,fb, the mass in the differentchemical species mp,X,fb, and the heating rate due to Lymancontinuum radiation Lyc,p) during a hydro time-step dt iscomputed individually for every stellar particle accordingto its properties. To obtain s,FB, the feedback is mappedto the hydro-mesh using a cloud in cell (CIC) scheme andtime-averaged over the time step dt. The SNe ejecta mass iscomputed by

mp,fb = mpi

ttpc+dtttpc

mdNdm

dmdt

dt, (31)

where m = m(Zp, t) is the initial mass of an individual

star that goes SN at an age of t (Raiteri et al. 1996). mp,fbis removed from the particle mass: mp = mp mp,fb. Themetal load of the ejecta is given by mp,Z,fb = mp,fb(Zp + ),where is the fraction of metals produced in the dying stars,the other species scale linearly with metallicity. The totalLyc heating rate is Lyc = Lyc,m + Lyc,pm. A fractionof energy load of the ejecta eSN

st

FB

is deposited into E(via e) and K, (1 SN) and SN, respectively. However,this does not account for the kinetic energy of the ejectathat must also be transferred to the hydro-mesh along withtheir mass:

(sEkin)

t

FB

=p

u2s,p2

st

FB,p

. (32)

Here we sum over all local contributions to kinetic energyfrom every individual particles p The momentum of theejecta transferred to bulk momentum of the gas changes thebulk kinetic energy of the gas by

(Ekin)

t

FB

=u

2(2us u)

st

FB

. (33)

To conserve energy, we add the difference to K

(K)

t

exFB

=(Ekin)

t

FB

(Ekin)t

FB

. (34)

The heating of the gas to very high temperatures by SNe isdescribed by a hot phase density h, obeying the conserva-tion equation

ht

+ (hu) + min [h u, 0]

=st

FB

h exp (t/h) ,(35)

where min [h u, 0] describes the loss of thermal energydue to adiabatic expansion2, and h exp(t/h) the decayof h due to successive mixing of the hot gas into the ISM.The half-lifetime-scale h/ log(2) is defined such that a SNe

2 Employing a ceiling (h u)max = 0 prevents producing hotphase when gas is compressed.

bubble shell at typical expansion velocity (roughly the speedof sound in the hot phase) travels roughly 1 kpc during thatperiod, which leads to h 1 Myr. eh 0.1 eSN is theconstant specific thermal energy of the hot phase gas.For consistency, the input parameters w and ew to thederivation of the radiative cooling rate radw (equation 16)are computed as follows:

w =w h,

ew =wew heh

w.

(36)

3 SIMULATIONS

3.1 Initial conditions

In the simulation domain with a volume of 0.5 Mpc3 we ini-tialize an isothermal gaseous disc with an exponential sur-face density profile residing in a static dark matter halo us-ing the potential-method of Wang et al. (2010), which givesinitial conditions similar to Agertz et al. (2009). The choiceof this setup is advantageous compared to a setup using aconstant vertical scale height of the disc alike that by Tasker& Tan (2009), because it is adiabatically stable.

In the absence of a stellar component, the exponentialgaseous disc is defined by its mass Mgas = 10

10 M, itsorientation of the disc angular momentum ngas assuminga radial scale length rgas = 3.5 kpc, an initially uniformmetallicity Zgas = 0.1 Z and a temperature Tgas = 4 104 K of the disc.

The dark matter is modelled by a static halo with aNFW-shaped density profile (Navarro et al. 1997). It onlycontributes to the dynamics via its gravitational acceleration

gdm =GMdmr

log(1+cdm) cdm1+cdm

(log(rs)

r3 cdmrdmr2rs

), (37)

at a given position with distance vector r from the halo cen-tre, its absolute value r and the scaled dimensionless radiusrs = (1 + rcdm/rdm). The NFW profile used is fully charac-terized by the halo mass Mdm = 10

12 M, the virial radiusrdm = 213 kpc, and a concentration parameter cdm = 12.

3.2 Individual runs

We performed eight isolated galaxy runs with model pa-rameters listed in Table 2. All runs were carried out with aroot grid of 2563 cells. AMR levels with a factor of 2 spa-tial and temporal refinement were created using refinementcriteria based on density; specifically, any cells with densityabove the minimum value of 0.01 M/pc

3 were tagged forrefinement up unto a specified maximum number of levels.In runs ref, nE, nB, nEnB, sSF, and sSF2 a spatial resolu-tion of 30 pc was obtained using six levels of refinement.The effects of the feedback implementation are explored withthe runs nE, nB, and nEnB in comparison with run ref, thatfeatures MIST with the reference parameters as given in Ta-ble 1.The runs sSF and sSF2 run feature a simplified model forthe ISM without phase separation and SN = 0. A thresh-old controlled star formation recipe is applied here, ac-cording to which stars are allowed to form at a free falltime efficiency of sSF = 0.01, if the local density exceeds

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8 H. Braun et al.

Table 2. Simulation runs.

ID LES ISM SN h Stop time

ref 30 pc Yes MIST 0.085 Yes 1.0 Gyr

nE 30 pc Yes MIST 0.0 Yes 1.0 GyrnB 30 pc Yes MIST 0.085 No 0.5 Gyr

nBnE 30 pc Yes MIST 0.0 No 0.4 Gyr

sSF 30 pc Yes Simple 0.0 Yes 0.8 GyrsSF2 30 pc No Simple 0.0 Yes 0.4 Gyr

lres5 60 pc Yes MIST 0.085 Yes 1.0 Gyr

lres4 120 pc Yes MIST 0.085 Yes 1.0 Gyr

sSF,min = 50mH cm3 and the local temperature is lower

than TsSF,max = 1.5 104 K. While in sSF2 the whole SGS-turbulence model is switched off, the model is still active insSF, but star formation is decoupled from K and the SGS-energy production terms related to MIST are switched offin this case (i.e. TI 0 and SN 0). Stellar particlesare not created as pairs (see Section 2.3.1) in sSF and sSF2,since `c is not defined in both cases.In addition two runs with fewer refinement levels were per-formed in order to investigate the effects of numerical reso-lution on the results of our simulations. The runs lres5 andlres4 feature effective resolutions of 60 pc and 120 pcusing five and four levels of refinement, respectively.Up to densities around 0.3 M pc

3 the resolution require-ment by Truelove et al. (1997) is easily satisfied in all runswith MIST. This limit can be shifted towards much higherdensities, if we consider the effective pressure instead of ther-mal pressure only. The Jeans length of the few densest cellsmay temporarily drop below 4 though, but never below thesize of a cell , before the dense region is disrupted againby feedback. The latter statements are not true in case ofsSF and sSF2, in which the Jeans length criterion is alwaysviolated in the dense clusters.The combined usage of LES and MIST in a simulation in-creases the amount of computational resources required byless than 10 per cent compared to runs without, and signifi-cantly less than an additional level of refinement (more than100 per cent).The isolated galaxies were evolved for at least one orbitaltime at 10 kpc radius from the centre ( 400 Myr).

4 RESULTS

4.1 Disk evolution in the ref run

Initially the disc is adiabatically stable, but as the gas isallowed to cool by radiation, it loses its thermal supportin height and collapses into a thin cold disc. The discbecomes Toomre-unstable and fragments into clumps. Inthose clumps the gas eventually becomes dense enough tobecome molecular and consequently begins to form stars.The SNe feedback of the newly formed stars then eventuallydisperses the clumps. A fraction of the stars formed in someof those clumps may form a stellar cluster that survives overa much longer time than the lifetime of about 20 Myr ofthe gas clump from which it originated. The stellar clusterstend to move towards the centre of the disc as a result

of dynamical friction, where they eventually merge into acentral agglomeration of stars, if they are not disruptedbefore by the tidal forces in the disc. However, they do notbuild up a bulge as their velocity dispersion is too small.The majority of the stars form a rather smooth disc witha scale height of a few hundred parsec. The structure ofthe stellar disc is shown in the top left panel of Fig. 1. Thestars stripped in the potential of the disc form tidal tailsaround their birth cluster.

The SNe also carve holes into the disc and launcha wind leaving the disc. The interplay between cooling,gravity, and SNe shapes the gaseous component into a fluffydisc, with holes and knots, as demonstrated in the top andbottom panels in the middle of Fig. 1. The wind mostlynot only consists of metal enriched hot gas from SNe, butalso carries a fraction of the original cold clump with it (seebottom left panel of Fig. 1). The wind originates from thecavities of the disc caused by the SNe at speeds rangingfrom 300 to 1000 km s1, and pushes a shell of cold orwarm gas outward. The ejected gas is either mixed into thewind, or falls back into the disc. Far away from its origin,the wind from all sources merges into a hot, but dilute,sub-sonically turbulent, and metal-rich ( Z) bubble thatcontinues to expand. The bottom panels of Fig. 1 give animpression of the metallicity, mass, and velocity structuresin the winds above the disc.

As seen in the top right panel of Fig. 1, active starformation occurs only in a few compact regions away fromthe centre. Because of the local metal enrichment due to SNeejecta, the threshold density for star formation drops. Thiscauses clumps to form stars earlier during their collapse,when the density is still relatively low, which lowers theirstar formation rate and makes them more prone to dispersalby SNe. The residual stellar clusters are fewer, lighter, andmore easily disrupted in the galaxys potential.The global star formation rate in the ref run is plotted asblack line in Fig. 2. Initially the amount of star formationincreases quickly, as the region of star formation grows. Itreaches its peak around 300 Myr after start of simulation,and then gradually declines due the consumption and themetal enrichment of the gas reservoir in the inner disc. Atthis stage approximately 30 per cent of the initial gas masshas been converted into stars.

4.1.1 Differences in the nE run

In the nE run we set SN = 0. Thus SNe feedback doesnot directly increase the unresolved turbulent energy K (seeequation 5). This has basically two effects on the overall evo-lution. On the one hand, star formation in a clump is activeover a longer period of time because of the higher star forma-tion efficiency in moderately turbulent clumps (Mc 10),causing more stars to form before a clump of gas is dis-persed. This leads to a slightly higher global star formationrate than in the ref run (see the blue line in Fig. 2). On theother hand, stars forming in a clump after the first stars havealready produced SNe have a significantly reduced velocitydispersion compared to their analogues from the ref run. Asa consequence, those stars are more likely to stay near theclump of their origin. In this case, stellar clusters tend to

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101 100 101 102

s [M/pc2 ]

101 100 101

HI [M/pc2 ]

102 101 100

H2 [M/pc2 ]

Z/5 Z 5ZZ [1]

106 104 102

[M/pc3 ]

400 200 0 200 400upolar [km/s]

Figure 1. The top row of panels shows the following projected quantities in the central (30kpc 30kpc) region of the galactic disc ofthe ref run after 1 Gyr (left to right): stellar column density s, HI column density HI, star formation column density SF. The bottomrow shows polar (100kpc 100kpc) slices of the central region of the galactic disc of the ref run at 1 Gyr (left to right): metallicity Z,total gas density , velocity perpendicular to the disc plane vpolar. The tick marks on the plot edges have a spacing of 5 kpc.

Figure 2. Global star formation rate MSF over simulation timefor the different runs ref, nB, nE, and nEnB in black, red, blue,

and green, respectively.

be more massive and more strongly gravitationally bound,and hence, their tidal tails are less prominent. Clusters areinitially more abundant and have a longer lifetime. Even-tually they merge into a few very massive clusters, whichaccrete gas, and host intermittent star formation, as the gasis driven apart due to feedback. They remain gravitationallybound and stable, as their stellar mass is sufficiently largeand concentrated.As a consequence, the resulting stellar and gaseous discsare more clumpy than in the ref run. This can be seenby comparing the projections of total gaseous density andstellar density in Fig. 3, where both stellar and gaseousclumps are fewer in numbers of appearance but more mas-sive with higher central densities. During the simulation ap-proximately 40 per cent of the initial gas mass was consumedby star formation after 1 Gyr.

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10 H. Braun et al.

ref

nE

nB

nEnB

101 102

total gas [M/pc2 ]

101 100 101 102

stellar s [M/pc2 ]

time : 400 Myr | field of view 1515 pcFigure 3. Comparison of total gas surface density in the left

column and the stellar surface density s in the right column be-

tween the different runs (from top to bottom: ref, nE, nB, nEnB)400 Myr after start of simulation.

4.1.2 Differences in the nB run

In the nB run we turned off the treatment of the hot SNejecta phase, i.e. the hot SN gas is directly mixed into thewarm phase. This allows the gas to radiate away the feed-back energy more quickly. Thus star formation in a clumpgoes on for a longer time, consuming a larger fraction of thegaseous mass. Once almost all gas is depleted the feedback

takes the lead, and drives stronger and faster winds than inthe ref run. Like in the nE run, the resulting stellar clus-ters are more stable and massive, and subsequently mergeto form very massive clusters, that cannot be disrupted inthe tidal field of the disc. Those massive clusters are host-ing continuous star formation. They keep accreting gas fromthe disc and converting it into stars, thereby sustaining theirgaseous mass at the same level for a long time.As in the nE run, the resulting gaseous and stellar discsare clumpy, but the stellar clusters are much more mas-sive, and the gas is depleted on much shorter time-scales(see Fig. 3). The stellar density in the centre of the clustersreaches values much greater than 103 M pc

3. This is whywe stopped this simulation after 0.5 Gyr. During this periodof time about 30 per cent of gaseous mass was turned intostars.

4.1.3 Differences in the nEnB run

Setting SN = 0 and turning off the treatment of the hot SNejecta phase combines the effects described above. This leadsto an even more violent evolution with a few very massivestellar clusters in the inner regions, while the gas is depletedquickly (see Fig. 3). We stopped this run even earlier thanthe nB run. The amount of gas consumed by star formationwas about 30 per cent in roughly 400 Myr.

4.2 Star formation

4.2.1 Time scales

The 20 Myr time-scale related to single star-forming re-gions reflects the period of time needed to produce enoughstars, such that their combined feedback is able to quenchstar formation, and to evaporate the most dense, centralregion of the star-forming cloud. If the environment is stilldense enough, star formation may continue in an compressedlayer around the expanding bubble. Depending on the massof the clump and its surroundings, star formation can con-tinue in this mode. A time series depicting the evolution ofone of those clumps is shown in Fig. 4. The impact of themost massive clumps is seen in the quick variations of theglobal star formation rate MSF on time-scales around 10 to40 Myr (see Fig. 5). Metal-enriched clouds tend to have ashorter lifetime compared to metal-poor clouds, as star for-mation can start at lower densities, because shielding fromradiation is enhanced due to higher dust abundances.

By assuming different threshold densities thr,i ={0.0, 0.01, 0.1, 0.32, 1.0} M pc3 to compute the reservoirof available gas, we derive global gas depletion time-scales

dep,i(t) =

(t)>thr,i

3SF3

=Mgas,i

MSF. (38)

To filter out quick variations, we apply a moving average

dep,i(t) =1

tav

t+tav/2ttav/2

dep,i(t)dt (39)

with tav = 50 Myr. Plots of the resulting depletion timesdep,i are shown in Fig. 5 for different choices of thr,i. Thetypical time-scales range from less than 100 Myr for thehighest threshold of 1 M pc

3 to a few Gyr for the totalgas content of the simulation domain. Gas above densities

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475 Myr

485 Myr

495 Myr

505 Myr

515 Myr

525 Myr

100 101 102 101 102

s [Mpc2 ] [Mpc

2 ]

SF>102 M

kpc2 yr

nLyc>106 cm2 s1

FB>5105M

kpc2 yr

H2 > 101 M

pc2

HI > 5M

pc2

field of view : (2.0 2.0) kpc2

Figure 4. Evolution of a gas clump from shortly before star

formation begins until dispersal. In the left panels the stellar sur-face density s in grey-scales is over-plotted with contours of the

surface density of the SNe feedback FB in red (indicating stel-

lar populations at ages between 4 and 40 Myr), the surfacedensity of Lyman continuum emission nLyc in green (indicating

stellar populations younger than 2 Myr), and the star formationsurface density SF in blue. Right panels show the total gas sur-face density in grey-scales over-plotted with contours of surface

density of cold gas HI in red, and the surface density of densemolecular gas H2 in blue.

Figure 5. Global gas depletion time-scale dep versus time in the

ref run. We plot moving averages of dep to eliminate short periodvariations caused by single star-forming regions. dep was derived

with respect to gaseous mass in regions with > thr. Blue, green,

orange, red, and purple lines show dep using threshold densitiesthr = {0.0, 0.01, 0.1, 0.32, 1} M pc3, or in terms of numberdensity nthr ' {0.0, 0.33, 3.3, 10, 33} cm3.

thr = 1 M pc3 is most likely actively star-forming. Con-

trary to the other cases, dep for thr = 1 M pc3 slightly

shrinks with time. Because of the increasing metallicity thethreshold density needed for star formation drops gradually.Star formation and subsequent feedback then prevent gasfrom becoming as dense as in metal-poorer environments.dep for thr = 0.32 M pc

3 remains almost constant at0.25 Gyr after the initial transient phase. For lower thr, depgrows with time, and the growth rate tends to increase withdecreasing thr. The growth of dep if thr 6 0.32 M pc3

is also related to the increase of metallicity. In metal-richenvironments gas forms stars already at lower densities, im-plying greater ff,c, but the efficiency PN (see equation 17)remains about the same. For this reason MSF is effectivelylowered faster than gas supply Mgas shrinks. We expect thatall dep will saturate, as the impact of increasing metallic-ity is further reduced in already enriched gas (see BS12).However, the magnitude of the depletion time-scales is wellwithin range of observationally inferred ones (Daddi et al.2010; Genzel et al. 2010).

4.2.2 Local efficiency

The star formation efficiency is defined by

SF =ff

ff, (40)

where a constant value for ff ' 0.01 is commonly assumed.In our simulations ff is computed dynamically from the lo-cal turbulent hydrodynamical state. While the star forma-tion efficiency in the cold phase, PN (see equation (17)), isalmost constant around PN ' 0.1 for all star-formingregions, the shielded molecular content fH2c and c,pa varysignificantly, such that ff is boosted in high density regions.As a consequence, a large contribution to the global star for-mation rate comes from just a few temporarily very activespots in the disc, which explains the relatively large varia-tions on short time-scales. As shown in Fig.6, we find a goodcorrelation ff

SF, which holds for all times in the ref

run and the nE run. In the other runs without treatment of

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103 102 101 100 101

SF [M/(kpc3 yr)]

103

102

101

ff[1

]

103 102 101 100 101

SF [M/kpc3 yr]

103 102 101 100 101

SF [M/(kpc3 yr)]

103

102

101

ff[1

]

10-4 10-3

Volume V [kpc3 ]

ref 200 Myr

ref 600 Myr

ref 1000 Myr

ref 400 Myr

nE 400 Myr

nB 400 Myr

nEnB 400 Myr

ref 1000 Myr

Figure 6. Star formation efficiency ff over star formation density SF. The left panel shows a two dimensional area weighted histogram

of the ref-data after 1 Gyr. The middle panel shows a time series (blue: 200 Myr, green: 600 Myr, black: 1 Gyr) of the mean ffcorresponding to a specific value of SF, while on the right panel a comparison of the latter between the individual runs (black: ref,

green: nE, blue: nB, red: nEnB) at 400 Myr is plotted.

a hot phase, the power-law slope flattens towards large SF,which is clearly a sign that the effectiveness of thermal feed-back plays a major role. The typical value of ff ' 0.01 is, onaverage, reproduced in runs with hot phase treatment, whilethose without produce stars at significantly higher averageefficiency.

4.2.3 H2 and HI

We find a very tight correlation between the star formationcolumn density SF and the molecular column density H2 ,as shown on the left panel of Fig. 7. This correlation impliesa robust power-law relation between SF and H2 withan exponent H2,SF = 1.05 0.06 slightly above one. Thiscorresponds to an almost constant depletion time of themolecular gas in star-forming regions around 80 Myr. Thiscorrelation holds for all runs independent of the simulationtime, as shown in the middle and right panels of Fig. 7.

The existence of the correlation itself is not surprising,as it is assumed in the MIST model equations that SF isproportional to fH2 (see equation 17). H2,SF ' 1, how-ever, is not imposed. Assuming a constant star formationefficiency PN, one would expect H2,SF ' 1.5, but in ourmodel PN and fH2 depend implicitly and non-linearly onc,pa, Mc, and lc, which are in turn nonlinear functionsof , c, w, K, and ew. BS12 already showed that acorrelation with H2,SF ' 1 can be found in the equilibriumsolutions of the MIST model equations. This suggests thatthe interplay of the implicit dependencies effectively resultsin an almost linear relation between estimated H2-massavailable to star formation and the star formation rate,which is an important feature of the self-regulating mech-anisms implemented in MIST. The depletion time-scaleof molecular gas from the equilibrium solutions (BS12)is about 1-2 Gyr, in agreement with that found by e.g.

Bigiel et al. (2011) from CO-observations. We derive anaverage depletion time of 80 Myr in our simulations, orshorter ( 40 Myr) in regions of higher molecular density.This is considerably, by a factor of around 25, shorter (seeFig. 7). But this time-scale matches the depletion timeof molecular gas within star-forming molecular clouds,found by Murray (2011)3, and that for dense moleculargas inferred from HCN-observations by Gao & Solomon(2004)4, and the total gas depletion time-scale dep forthr = 1 M pc

3. This discrepancy between the depletiontime-scales of molecular gas is caused by the fact that weestimate the amount of gas involved in the processes ofactive star formation by calculating the molecular fractionof gas in the centres of cold cloud complexes in equilibriumwith an external radiation field. However, there are alsoenvironments containing significant amounts of H2 that arenot in equilibrium with radiation or only partially shielded.Molecular material that is transported from shielded tonot shielded regions by turbulent motions or SNe blastwaves without being instantaneously dissociated is nottracked in our model for example. This material, as wellas gas that is partially molecular but not dense enoughto form stars, contributes to observed molecular hydrogencolumn densities, but is not present in our simulations.Our data on the H2-content of the ISM do reflect theamount of dense molecular gas that is actively formingstars, but do not reproduce the full amount of molecular

3 Murray (2011) find a GMC star-forming efficiency GMC ' 0.12and related free fall times ff,GMC ' 10 Myr. A combination ofboth yields a depletion time-scale of dep,GMC ' 80 Myr.4 Gao & Solomon (2004) find a linear correlation between

the galactic star formation rate SF,gal and the amountMdense of what they call dense gas in a galaxy: SF,gal =

1.8Mdense/108yr1. This corresponds to a dense gas depletion

time of 60 Myr.

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103 102 101 100 101

H2 [M/pc2 ]

105

104

103

102

101

100

101

SF

[M/(k

pc2

yr)

]

103 102 101 100 101

H2 [M/pc2 ]

103 102 101 100 101

H2 [M/pc2 ]

105

104

103

102

101

100

101

SF

[M/(k

pc2

yr)

]

dep = 320Myr

dep = 80Myr

dep = 20Myr103 104

Area A [pc2 ]

ref 200 Myr

ref 600 Myr

ref 1000 Myr

ref 400 Myr

nE 400 Myr

nB 400 Myr

nEnB 400 Myr

ref 1000 Myr

Figure 7. Star formation column density SF over H2 column density H2 . Plots and colours are arranged as in Fig. 6. Additionally, in

the left panel the dotted, dash-dotted, and dashed green lines indicate constant depletion time-scales of 20, 80, and 320 Myr, respectively.

100 101 102 103

HI+H2 [M/pc2 ]

105

104

103

102

101

100

101

SF

[M/(

kpc2

yr)

]

100 101 102 103

HI+H2 [M/pc2 ]

100 101 102 103

HI+H2 [M/pc2 ]

105

104

103

102

101

100

101

SF

[M/(

kpc2

yr)

]

2HI+H21.5HI+H21HI+H2

103 104

Area A [pc2 ]

ref 200 Myr

ref 600 Myr

ref 1000 Myr

ref 400 Myr

nE 400 Myr

nB 400 Myr

nEnB 400 Myr

ref 1000 Myr

Figure 8. Star formation column density SF over HI + H2 column density HI+H2 . Plots and colours are arranged as in Fig. 6. FitsSF 2HI+H2 , SF

1.5HI+H2

, and SF 1HI+H2 are shown as dashed cyan, dashed blue, and dashed green line, respectively.

gas seen in CO-observations. Coarse graining our H2-datachanges neither the inferred depletion time-scales nor theslope H2,SF significantly, which is consistent with Gao &Solomon (2004), who used galactic quantities.Without following the chemical evolution of molecular andatomic gas, we cannot, unfortunately, distinguish betweenatomic gas and gas that would observationally be consideredmolecular. So we focus on the combined HI + H2 content.The general shape of SF over HI + H2 shown in the leftpanel of Fig. 8 is similar to observational relations(e.g. inSchruba et al. 2010), but our distribution is shifted towardshigher densities and rates.As demonstrated by Schruba et al. (2010), the distributionin SF-HI+H2 space changes from low to high HI+H2between the vertical barrier behaviour in SF-HI space

and the linear relation observed for H2. This transitionhappens around the critical density for atomic to molecularconversion, which is metallicity dependent. Increasing thedensity of gas only leads to an increase in the moleculardensity, if its density was already above the critical density.Because of that the atomic density saturates. This explainsthe vertical barrier-shaped distribution in SF-HI space,if we take into account that the amount of surroundingatomic gas does not affect star formation in the moleculargas.From the high-density tail in the left panel of Fig. 8 onewould rather infer a power-law slope SF,HI+H2 ' 1.5 thana linear relation with SF,HI+H2 ' 1, but also a power-lawslope SF,HI+H2 ' 2 seems possible. At this point wecannot distinguish whether the reason for this is either that

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101 102

HI+H2 [M/pc2 ]

104

103

102

101

100

101

SF[M

/(kpc2

yr)]

ref 30 pc

ref 60 pc

ref 120 pc

ref 240 pc

ref 480 pc

ref 960 pc

ref 1920 pc

ref 1000 Myr

Figure 9. Star formation column density SF over HI + H2column density HI+H2 for different resolution scales crse =

{30, 60, 120, 240, 480, 960, 1920} pc as solid lines. The full res-olution data from the ref run were coarsened by the appropriatepowers of two. Additionally, data from 10 consecutive root grid

time steps (separated by dt ' 0.65 Myr) around a simulation timeof 1 Gyr were averaged to reduce scatter.

stellar feedback prevents the high density tail from beingpopulated up to densities at which SF,HI+H2 ' 1 couldbe observed, or the super-linear relation also seen in theequilibrium solutions of BS12 is recovered. With increasingmetallicity of the star-forming gas the critical density of theatomic to molecular transition drops gradually. This effectis demonstrated by the shift towards lower HI+H2 withincreasing time in the middle panel of Fig. 8. A linear fit tothe tail of the SFHI+H2 distribution in ref after 1 Gyryields a gas depletion time dep,HI+H2 ' 400 Myr, roughlyconsistent with dep(0.32 M pc

3 > thr > 0.1 M pc3)

(see Section 4.2.1). In the ref run roughly the same amountof stars and metals were produced after 1 Gyr as in thenEnB after 0.4 Gyr. Nevertheless the SFHI+H2 relationsare very different. While the high-density tail indicatingthe transition from HI to H2 becomes shallower and moreprominent in the course of the disc evolution, the shape ofSF over HI+H2 in the nEnB run at 0.4 Gyr is about thesame as in the early stages of ref. This is a consequenceof inefficient mixing of the hot, metal-rich material fromSNe with cold dense, but still metal-poor material thatpossibly could be turned into stars. The shift betweenthe SFHI+H2 relations for different runs in the rightpanel of Fig. 8 is mainly caused by gas consumption, asthe produced metals have not been mixed into the star-forming material yet. So, the shift between our results andobservational findings is, on the one hand, a consequence ofthe lower metallicity and higher gas contents, compared toobserved local galaxies.

On the other hand the resolution scale crse, at whichthe relation between star formation and gas density is

evaluated, influences the critical density of the atomic tomolecular transition too, as demonstrated in Fig. 9. Oncethe averaging scale is considerably larger than a typicalstar-forming region, i.e. above 120 pc in the ref run, thetransition range in SFHI+H2 space seems to be shiftedtowards lower densities. Observational data like in Bigielet al. (2011) usually correspond to averages over muchlarger areas, as observations used for statistical analysisof star formation in disc galaxies have spatial resolutionsabout some 102 pc, compared to our 30 pc. The starformation rate of a given active region is put into relationwith a larger volume that contains besides the densestar-forming gas large amounts of ambient, inactive gas.Since the tracers have a certain lifetime, observationallymeasured star formation rates are temporal averages aswell, while our data follow the instantaneous star formationrate. However, the shift of the position of the knee in theSFHI+H2 space with coarsening is a combination ofresolution effects and the already discussed effects of stellarto gaseous mass ratio and the metallicity in the dense gasof the discs. Within a resolution element the fraction of gasthat is not directly involved in the star formation processincreases with decreasing resolution due to averaging.The high density tail in SF-HI+H2 space indicating acorrelation of star formation with gas density becomestherefore less pronounced in case of coarser resolution.

4.2.4 Turbulence

The main factor that determines the star formation rateSF is the estimated amount of shielded H2. The molecularfraction of the cold phase fH2 is computed by finding thelocation of the photo-dissociation front in a spherical clumpof size lc and density c,pa. The turbulent state has a majorimpact on these quantities via two competing effects. First,turbulent motions boost the production rate of H2 vialocal density enhancements of the cold phase. Secondly theturbulent contribution dominates the effective pressure inthe cold phase and may play a significant role in the warm-phase pressure as well. For higher K the difference betweenthe phase densities c,pa > and w,pa 6 becomes smaller,which partially counteracts the boost of the productionrate. The interplay between these processes determinesthe minimum K for star formation. In the left panel ofFig. 10 one can see that in the ref run star formation isstrongly suppressed for K > 103 km2 s2. For high K alsothe impact of feedback becomes important. Since feedbackcauses the gas to expand, however, there is very little high-density gas that is also strongly turbulent. This explains therelatively narrow range around K ' 100 km2s2 in whichstar formation actually occurs, which corresponds to avelocity dispersion around ' 10 km s1 or a turbulent RMSMach number of about 10, consistent with observations(Shetty et al. 2012).

The enrichment of the gas with metals lowers theminimum K for which gas of a given density can becomemolecular. This effect can be seen in the middle panel ofFig. 10. The degree of enrichment differs throughout thedisc, and hence, star formation is possible for a broader

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100 101 102 103

K [km2 /s2 ]

104

103

102

101

ff[1

]

100 101 102 103

K [km2 /s2 ]

100 101 102 103

K [km2 /s2 ]

104

103

102

101

ff[1

]

105 104 103

Volume V [kpc3 ]

ref 200 Myr

ref 600 Myr

ref 1000 Myr

ref 400 Myr

nE 400 Myr

nB 400 Myr

nEnB 400 Myr

ref 1000 Myr

Figure 10. Star formation efficiency ff over K. Left panel: phase plot for the central region of the disc in the ref run. Middle and right

panels: occupied volume Vbin > 3104 kpc3 contours of phase diagrams at different times in the ref run, and at 0.4 Gyr in the differentruns. Plots and colours are arranged as in Fig. 6.

range of K.

The evolution of an individual star-forming region iseasily understood by following its path in the ffK diagram.Starting at the minimal K for star formation, accretion, col-lapse, Lyman-feedback, and phase separation rapidly pro-duce turbulence until the production is balanced by dis-sipation. Simultaneously SF and ff increase and eventu-ally reach a self-regulated state. This regime is associatedwith the most densely populated area in ff -K space aroundK ' 102 km2 s2. But once SNe begin to dominate the ther-mal evolution locally, density, ff , and SF reached their peakvalues and subsequently decline. Comparing the contours ofthe runs with SN = 0 to those with SN = 0.085 in Fig. 10,it appears that stars are formed in more turbulent envi-ronments in the latter case. This reflects the enhancementof K by SN feedback. Switching the hot phase treatmentoff reduces the impact of thermal feedback, as all thermalSN feedback energy is instantly mixed into the dense warmgas. Since much stronger feedback is needed to dilute thegas, the gaseous disc is clumpier and the clumps tend to bedenser in this case. This increases both the star formationefficiency and K as shown in the right panel of Fig. 10. Oncethe feedback begins to dominate in a clump, the subsequentexpansion is much faster without hot phase treatment, andhence, star formation with low ff in strongly turbulent gasis cut off.

4.3 Drivers of turbulence

In the MIST model implementation we consider two groupsof sources of SGS turbulence. First there is the production ofsmall-scale turbulence by large-scale shear and compression.The corresponding source term SGS in equation (10) maybe both positive or negative, indicating whether small-scaleturbulence is driven by resolved motions (direct cascade) or

101 100 101 102 103 104

K [km2 /s2 ]

107

106

x/(K)[yr

1]

int/(K)

SGS/(K)

SGS/(K)

tot/(K)

tot/(K)

Figure 11. Volume-weighted mean of inverse production time-

scales of turbulent sub-grid energy /(K) over specific unre-solved turbulent energy K in the star-forming part of the disc

from the ref run after 1 Gyr. The inverse time-scale SGS/(K)of production by resolved motions via the turbulent stress tensoris shown in blue. The inverse time-scale int/(K) := (TI +SN)/(K) of the non-adiabatic MIST-sources is printed in red,

and the inverse time-scale tot/(K) := (int + SGS)/(K) ofthe total small scale turbulent energy production. A solid line

indicates positive, and a dashed line negative values.

the other way around (inverse cascade). The internal sources

int := SN + TI (41)

are specific to MIST. The physical processes modelledhere are the phase separation due to thermal instability(TI, equation (21)), and small-scale motions caused bySNe bubbles and instabilities in their blast waves (SN,

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16 H. Braun et al.

101 100 101 102 103 104

K [km2 /s2 ]

107

106

x/(K)[yr

1]

int/(K)

SGS/(K)

SGS/(K)

tot/(K)

tot/(K)

Figure 12. Volume weighted mean of inverse production time-scales of turbulent sub-grid energy /(K) over specific unre-

solved turbulent energy K in dense ( > 0.032 M pc3) areasof the disc from the ref run after 1 Gyr. Line colours and stylesare arranged as in Fig. 11.

101 100 101 102 103 104

K [km2 /s2 ]

107

106

x/(K)[yr

1]

int/(K)

SGS/(K)

SGS/(K)

tot/(K)

tot/(K)

Figure 13. Volume weighted mean of inverse production time-

scales of turbulent sub-grid energy /(K) over specific unre-solved turbulent energy K in the star-forming part of the disc

from the nE run after 1 Gyr. Line colours and styles are arrangedas in Fig. 11.

equation (22)). Our model enables us to disentangle the con-tributions to the total production rate tot := SGS + intfrom the turbulent cascade and internal sources to answerthe question, which processes are most relevant in whichregime.

We can distinguish four regimes, as demonstrated inFig. 11 for the ref run:

(i) K . 10 km2 s2: negative SGS indicate expand-

ing environments. In this case, it is the signature of in-falling dilute gas. The material found in this regime ismost likely to fall freely towards some dense clump. int .SGS is caused by the enforcement of the floor Kmin =0.05 km2 s2 ec.5

(ii) 10 km2 s2 . K . 100 km2 s2: turbulence ismainly supported through the turbulent cascade SGS. Asub-dominant contribution is TI. This regime is typical forstar-forming clumps in their early evolutionary stages.

(iii) 100 km2 s2 . K . 1000 km2 s2: SGS is the maindriver of unresolved turbulence in low density environments,but int dominates and SGS becomes negative in dense en-vironments (see Fig. 12). The latter case indicates an ex-panding environment. For greater K the internal sourcesint become increasingly dominated by SN. Star-formingregions in late evolutionary stages reside in this regime.

(iv) K & 1000 km2 s2: SNe feedback dominates not onlythe thermal evolution but also all other contributions to tur-bulence production. The gas in this regime is hot and rapidlyexpanding.

The power-law behaviour tot/(K) K0.5 for interme-diate K indicates that tot is, on average, balanced by tur-bulent dissipation SGS K1.5. While TI is slightly en-hanced in the absence of SN (i.e. in the nE and nEnB run),SGS remains roughly unchanged (see Fig. 13). In this casethe low K-regime is extended up to K . 50 km2 s2. Afurther consequence of SN = 0 is the lack of effective tur-bulence production for K > 103 km2 s2.

4.4 Resolution study

With decreasing resolution less of the dynamics in the discsis resolved. In particular the relative importance of thestellar feedback compared to gravity is reduced, as the feed-back energy is deposited in a larger volume. This leads tomore clumpy discs in the low-resolution runs (see Fig. 14).The resulting gaseous structures like knots and connectingfeatures are much more extended and more massive thanone would expect from the ratio of resolutions alone. BS12have shown that the size of the reference volume, whichcorresponds to a numerical resolution element, does notaffect the equilibrium solutions. However, the time-scalesrelated to an evolution from an arbitrary state towardsequilibrium in the BS12 model are considerably longer forlarger , which is caused by additional turbulent modes ina larger reference volume.This effect could be compensated for by adjusting SN, eh,and h. The physical reasoning is that an increased fractionof the energy released by SNe is present in the form ofmotions on larger spatial scales driven by the expandingSNe bubbles which provide additional pressure support.Despite the differences in the disc structure, the globalstar formation rates in the ref and lres5 run are in goodagreement with each other (see Fig. 15), although the initialtransient phase in the lres5 run ( ' 60 pc) lasts longerthan in the ref run ( ' 30 pc). In both runs about 30per cent of the initial mass were transformed into stars

5 A minimum level of turbulence is needed for the implementa-

tion of the SF11-model to work properly, as SGS depends di-rectly on the instant value of K (see Eqn. (10,8)).

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=

=

=

30 pc

60 pc

120 pc

101 102

total gas [M/pc2 ]

101 100 101 102

stellar s [M/pc2 ]

time : 1 Gyr | field of view 1515 pcFigure 14. Comparison of total gas surface density in the left

column and the stellar surface density s in the right column

between the runs of different effective resolution (from top tobottom: ref, lres5, lres4) 1 Gyr after start of simulation.

Figure 15. Global star formation rate MSF over simulation timefor the different runs lres4, lres5, and ref in green, orange, and

black, respectively.

102 101 100 101

H2 [M/pc2 ]

104

103

102

101

100

101

SF

[M/(

kpc2

yr)

]

30 pc

60 pc

120 pc

dep = 320Myr

dep = 80Myr

dep = 20Myr

Figure 16. Star formation column density SF over H2 column

density H2 for runs with different numerical resolution: lres4,lres5, and ref in green, orange, and black, respectively.

101 102

HI+H2 [M/pc2 ]

104

103

102

101

100

101

SF[M

/(kpc2

yr)]

30 pc

60 pc

120 pc

Figure 17. Star formation column density SF over HI + H2 col-

umn density HI+H2 for runs with different numerical resolution:

lres4, lres5, and ref in green, orange, and black, respectively.

after 1 Gyr. Star formation in the lres4 run ( ' 120 pc) isless efficient, such that only 15 per cent of the mass wasturned into stars after 1 Gyr. Due to lack of resolution andthe effect discussed above, the relevant structures for starformation either are not sufficiently resolved or simply donot form.Regardless of the differences between the runs, the star

formation versus density relations, as shown in Fig. 16 forSF over H2 and in Fig. 17 for SF over HI+H2 , stayroughly unchanged, which is a consequence of the internalregulation of MIST. The slight shift in the case of the plotswith = 120 pc in Figs 9 and 17 is caused by the generally

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full MIST

no MIST

no SGS no MIST

101 102

total gas [M/pc2 ]

101 100 101 102

stellar s [M/pc2 ]

time : 400 Myr | field of view 1515 pcFigure 18. Comparison of total gas surface density in the

left column and the stellar surface density s in the right column

between the runs using different ISM models (from top to bottom:ref, sSF, sSF2) 400 Myr after start of simulation.

lower star formation rate in the lres4 run compared to theref run.

4.5 Impact of subgrid scale model

Fig. 18 shows that the stellar discs in the runs without MIST(sSF and sSF2) are dominated by a few very massive stellarclusters after 400 Myr. The gas follows the distribution ofstars and is consequently concentrated in the stellar clus-ters. Star formation occurs only in those places, since thereis no high-density gas elsewhere. Clearly, the Truelove et al.(1997) criterion is not fulfilled within those clusters, as nei-ther SGS turbulence energy nor SNe energy feedback cansupport the gas sufficiently in both runs. The SGS model isnecessary to keep the gas (and in the consequence the stars)from clustering too much in order to obtain a realisticallysmooth and flocculent disc.

Figure 19. Global star formation rate MSF over simulation timefor the different runs ref, sSF, and sSF2 in black, purple, and

green, respectively.

The global star formation rate of a few M yr1 in sSF and

sSF2 is comparable to the ref run (see Fig. 19). This is aconsequence of the choice of the parameters in the simplifiedstar formation model (sSF = 0.01, sSF,min = 50mH cm

3

and TsSF,max = 1.5 104 K), which are chosen to match theaverage star formation properties in the runs with MIST.However, both runs with the simplified model lack the strongvariations on time-scales between 10 and 30 Myr seen in runswith MIST. As pointed out in Section 4.1 and 4.2.1, the vari-ations are related to the life cycle of individual star-formingregions. The SNe feedback disrupting the cold dense gas af-ter a phase of intense star formation limits the lifetime ofan active region in the MIST runs. The lifetime of these re-gions is not limited in the non-MIST runs, as the feedbackenergy is radiated away before it could affect the gas. Thisis known as the so-called over-cooling problem. Moreover,the stellar particles are not inserted with peculiar velocities,representing the unresolved motions of the star-forming gas(see Section 2.3.1). These differences in the treatment ofstar formation and feedback result in the formation of largestrongly bound clusters, in which continuous star formationis fueled by the accumulation of gas through accretion andmergers into even larger clusters. This is a runaway process.After an initial transient phase the global star formationrate reaches a plateau around 6 M yr

1 (see Fig. 19). Dueto the stabilizing effect of the SGS turbulence energy thetransient phase in sSF lasts 50 Myr longer than in sSF2.The gradual decline of the star formation rate seen in runswith MIST due to metal enrichment (see Section 4.2.1) doesnot occur in sSF and sSF2, as neither sSF nor sSF,min de-pend on metals. The KS relation, depicted in Fig. 20, showsa tight correlation with the expected slope of SF 1.5.

5 DISCUSSION AND CONCLUSIONS

In this paper we introduced MIST, which is a SGS modelbased on the semi-analytical BS12 model (Braun & Schmidt2012) for the turbulent multi-phase ISM. We implementedMIST into the code Nyx (Almgren et al. 2013) to run adap-tively refined LES of IDG with different stellar feedback pa-rameters. For the first time, a complete treatment of the nu-merically unresolved turbulence energy via the SGS model

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102 103

HI+H2 [M/pc2 ]

103

102

101

100

101

SF[M

/(kpc2

yr)]

full MIST

no MIST

no SGS no MIST

1.5HI+H2

Figure 20. Star formation column density SF over HI + H2column density HI+H2 for the runs ref, sSF, and sSF2 in black,

purple, and green, respectively.

of SF11 is applied in such simulations. The star formationrecipe follows Krumholz et al. (2009) and Padoan & Nord-lund (2011). In our fiducial galaxy model, supernova feed-back produces both SGS turbulence energy and heat. Sincethe injected heat would instantly cool away in dense en-vironments, we suppress cooling in the hot gas producedby feedback over a time-scale that is given by the typicalsize and expansion velocities of hot SN bubbles in the ISM.Comparison runs demonstrate that the following propertiesof star formation in disc galaxies are reproduced if both tur-bulent and thermal feedback are applied:

(i) an average total gas star formation efficiency ff ' 0.01(ii) ff is enhanced in dense environments(iii) a galactic depletion time-scale dep 0.3 . . . 1 Gyr in

a gas-rich galaxy(iv) a linear relationship between SF and H2(v) a lifespan of molecular clouds tGMC 10 . . . 30 Myr(vi) a velocity dispersion in star-forming regions SF '

10 km s1

We observe three modes of star formation in our simulations.First, stars are formed in the metal-poor gas that occurs inisolated clouds. In this material, the threshold density forstar formation is rather high. Therefore, the gas is quicklyturned into stars and the ensuing feedback disrupts the cloudviolently in the simulation with full feedback. Subsequentdispersion of the gas effectively quenches star formation,as the expanding shell is diluted below the threshold den-sity. Typically, a stellar cluster is left behind that surviveslonger than the original gas cloud. Eventually, the stars aredispersed in the discs potential or merge into more stableand massive clusters. With increasing amounts of metals inthe gas, a different mode of star formation emerges. Owingto the decreasing threshold density, active star formation ispossible in more extended regions of lower density. Although

star formation is still quenched by feedback in these regions,waves of star formation propagate through the metal-rich in-ner parts of the disc. Since the feedback is less violent, thewinds launched in this mode are slower. The third mode ofstar formation is hosted by massive stellar clusters that lo-cally dominate the gravitational potential. They are formedvia mergers of the clusters generated in metal-poor star-forming regions. Once they are massive enough, they beginto accrete gas from their surroundings, and consequentlyhost star formation. Depending on their mass and the cho-sen feedback parameters, star formation is continuous or in-termittent. The massive clusters are candidates for globu-lar cluster progenitors, although they continue to grow andslowly spiral towards the center in our isolated disc simu-lation, where they merge into the central stellar agglomer-ation. In a cosmological galaxy simulation, mergers couldinterrupt this process by kicking the clusters out of the discplane and thereby cutting them off of gas supply. Moreover,it is possible that late feedback from SNe of type Ia (Agertzet al. 2013) has an impact on the star formation induced byclusters. In the current implementation, stellar populationsolder than roughly 40 Myr are quiescent. Taking SNe Iainto account, the period of active feedback might increaseto a few 100 Myr. Although this type of feedback is muchless intense than the feedback caused by SNe II, it mightprevent the accretion of gas in dense stellar clusters. Alsoso-called early feedback (e.g. see Stinson et al. 2013) in theform of turbulent feedback due to stellar winds from mas-sive stars might make residual stellar clusters more prone todisruption, since the velocity dispersion of the stars wouldbe enhanced.

For all of our runs using MIST, we found a robust,almost linear relationship between shielded molecular gasand the star formation rate, which is a consequence of self-regulatory processes in MIST. This kind of relation also ap-pears in the semi-analytic model (BS12) and in observations(Bigiel et al. 2011). The related H2-consumption time-scaleof 40 to 80 Myr is in good agreement with observations ofactively star-forming clouds (Murray 2011). The global gasdepletion time-scale of about 0.5-1 Gyr, which is inferredfrom the mean star formation rate in the simulation, is inagreement with the observed gas depletion in gas-rich galax-ies (Daddi et al. 2010). The gradual increase of this time-scale suggests that depletion times of a few Gyr could bereached in later evolutionary stages, comparable to thoseobserved for local galaxies. The gas consumption time-scaleinferred from the relation between SF andHI+H2 is alsoin agreement with observations of gas-rich galaxies. Theseresults confirm our method of computing the star formationrate. However, the amount of molecular gas following fromour model is generally too low to be consistent with observa-tions. The missing H2 is a consequence of using a Stromgren-like ansatz to obtain an equilibrium solution instead of fol-lowing the chemical evolution. Although this allows us toreliably estimate the amount of molecular gas that residesin shielded areas, a large fraction of the H2 mass would befound in surrounding regions if detailed chemical reactionnetworks were computed in the cold and warm phases. Asthe star formation rates in our simulation reproduce obser-vations reasonably well, it appears that the shielding fromradiation, and hence the lack of heating, is the major factorthat controls star formation.

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20 H. Braun et al.

While the star formation efficiency PN with respectto the shielded gas is almost constant for all star-formingregions, regardless of density and metallicity, the efficiencyff with respect to the total gas density does vary dueto the impact of metals and density on the shielded H2content. This results in an enhanced efficiency ff in regionswith high star formation rate, contrary to the sometimesemployed constant efficiency parameters (e.g. Agertz et al.2013). Since the production of H2 strongly increases withthe density of the unresolved cold-gas clumps, which isenhanced by turbulence via the dependence of the cold-gasdensity PDF on the turbulent Mach number in our model,a minimal level of SGS turbulence energy is necessary forstar formation. Stellar feedback, which is associated withvery high SGS turbulence energy, reduces star formation.Consequently, star formation occurs only for an intermedi-ate range of SGS turbulence energy, centred around a peakvalue that roughly corresponds to a velocity dispersion of10 km s1, comparable to observed values Shetty et al.(2012). Apart from feedback, the turbulent cascade isan important source for building up the moderate levelof turbulence in star-forming regions before SN feedbackbecomes dominant.While Agertz et al. (2013) make efforts towards accountingfor all relevant feedback mechanisms - these are stellarwinds from massive stars, radiation pressure, SNe II,SNe Ia, and mass loss by mass-poor stars - in a realisticfashion by taking age and metallicity of a stellar populationinto account and using appropriate application channelsand schemes for each of them reflecting their physicalimpact on the ISM, their star formation recipe is relativelysimple. In contrast to their approach, we focus on a moreelaborate way to describe the sub-resolution structureand processes in the ISM. The non-thermal pressure -actually the trace of the turbulent stress tensor - helpsto stabilize gas against gravity particularly in cold anddense environments. The SGS-energy allows us to collectthe effect of subresolution-scale motions excited by SNe,which is our counterpart to the momentum feedback inAgertz et al. (2013). While over-cooling is usually avoidedby suppressing all cooling for period of more than 10 Myr,the combined effect of non-thermal pressure and turbulentfeedback reduces the need for delayed cooling in areas ofactive SNe feedback in our simulations.

MIST coupled to an SGS model is suitable for the use incosmological zoom-in simulations with effective resolutionsof about 10 to 100 pc without substantial modifications.However, the current model framework of MIST has to beadjusted to treat extremely metal-poor gas and the radiationbackground consistently. Apart from that the incorporationof reaction networks to track the actual molecular contentin the different gas phases and of additional stellar feedbackmechanisms like stellar winds from massive stars and SNe Iawill further enhance the model.

ACKNOWLEDGEMENTS

HB was financially supported by the CRC 963 of the Ger-man Research Council. The work of AA was supported bythe SciDAC FASTMath Institute, funded by the Scientific

Discovery through Advanced Computing (SciDAC) programfunded by U.S. Department of Energy. HB, WS, and JCN ac-knowledge financial support by the German Research Coun-cil for visits at LBNL. We thank Hsiang-Hsu Wang for dis-cussions on the initial conditions of idealized IDG. The sim-ulations presented in this article were performed on the Su-perMUC of the LRZ (project pr47bi) in Germany. We alsoacknowledge the yt toolkit by Turk et al. (2011) that wasused for our analysis of numerical data. We owe thanks tothe referee B. Robertson for a careful and helpful report thathelped us to improve this paper.

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