Interstellar Turbulence and Star Formation
aa r X i v : . [ a s t r o - ph . GA ] N ov Computational Star FormationProceedings IAU Symposium No. 270, 2010J. Alves, B. Elmegreen, J. M. Girart, & V. Trimble, eds. c (cid:13) Interstellar Turbulence and Star Formation
Alexei G. Kritsuk, Sergey D. Ustyugov, and Michael L. Norman UC San Diego, 9500 Gilman Drive MC 0424, La Jolla CA 92093-0424, USAemail: [email protected], [email protected] Keldysh Institute of Applied Mathematics, Miusskaya Sq. 4, 125047, Moscow, Russiaemail: [email protected]
Abstract.
We provide a brief overview of recent advances and outstanding issues in simulationsof interstellar turbulence, including isothermal models for interior structure of molecular cloudsand larger-scale multiphase models designed to simulate the formation of molecular clouds.We show how self-organization in highly compressible magnetized turbulence in the multiphaseISM can be exploited in simple numerical models to generate realistic initial conditions for starformation.
Keywords.
ISM: structure, ISM: clouds, ISM: magnetic fields, turbulence, methods: numerical
1. Introduction
Since most of the ISM is characterized by very large Reynolds numbers, turbulentmotions control the structure of nearly all temperature and density regimes in the inter-stellar gas (Elmegreen & Scalo 2004). Because of that, turbulence is often viewed as anorganizing agent forming and shaping hierarchical cloudy structures in the diffuse ISMand ultimately in star-forming molecular clouds (e.g., V´azquez-Semadeni & Passot 1999).Nonlinear advection dominating the dynamics of such highly compressible magnetizedmulticsale self-gravitating flows makes computer simulations practically the only toolto study fundamental aspects of interstellar turbulence, even though effective Reynoldsnumbers in numerical models are always limited by the available computational resource(e.g., Kritsuk et al. 2006).Over the last five years three-dimensional numerical simulations fostered the devel-opment of theoretical concepts concerning the interstellar medium undergoing nonlin-ear self-interaction and self-organization in galactic disks. One can conventionally dividethese models designed to tackle various aspects of interstellar turbulence into three differ-ent classes depending on the range of resolved scales and physics included: (i) mesoscale models that cover evolution of multiphase ISM in volumes with linear size of a few-to-tenkpc and resolve the flow structure down to a fraction of 1 pc (e.g., galactic fountain mod-els developed by de Avilles & Breitschwerdt 2005-07); (ii) sub-mesoscale models resolvingthe scale-height of the diffuse H i ( ∼
100 pc) and usually limited to only warm-to-coldneutral phases (WNM and CNM) (e.g., Kissmann et al. 2008; Gazol et al. 2009; Gazol& Kim 2010; Seifried et al. 2010); and (iii) microscale models for molecular cloud (MC)turbulence that assume an isothermal equation of state and deal with <
10 pc-sized sub-volumes within MCs.
Global galactic disk models (e.g., Tasker & Bryan 2006-08, Wada2008 and references therein) which represent the future of direct ISM turbulence mod-eling, are currently resolving scales down to ∼
10 pc, i.e. insufficient to properly followthe thermal structure of self-gravitating multiphase ISM.Mesoscale models of supernova-powered (SNe) galactic fountain have demonstrated theimportant role of dynamic pressure in the ISM that keeps large fractions of the gas massout of thermal equilibrium and elevates gas pressures of GMCs to the observed levels11920 Kritsuk, Ustyugov & Normaneven without direct action of self-gravity (Korpi et al. 1999; Mac Low et al. 2005; deAvilles & Breitschwerdt 2005-07; Joung et al. 2006-09). They also show that the effectiveintegral scale of the SN-driven turbulence ( ∼
75 pc) is about half the scale height of theH i gas in the inner Galaxy [100 −
150 pc (Malhotra 1995)] and outline a general pictureof probability distributions for the mass density, magnetic field strength, and thermalpressure in the turbulent ISM in disk-like galaxies.Supersonic isothermal turbulence simulations in periodic boxes representative of themicroscale models provided many important insights into the physics of interstellar turbu-lence and helped to guide the interpretation of observations. These numerical experimentshighlighted the importance of nonlinear advection as a major feature of compressible tur-bulence (Pouquet et al. 1991). At high Mach numbers, turbulent flows are dominated byshocks; therefore the velocity spectra are steeper than the Kolmogorov slope of − / − M s >
3, strong shock interactionsand associated nonlinear instabilities create sophisticated multiscale pattern of nested U -shaped structures in dynamically active regions morphologically similar to what isobserved in molecular clouds (Kritsuk et al. 2006a). Scaling of the first-order velocitystructure functions S ( δ u ) ∼ ℓ . (where δ u ( ℓ ) = u ( x ) − u ( x + ˆe ℓ ), S p ( δ u ) = h [ δ u ( ℓ )] p i and h . . . i indicates averaging over an ensemble of random point pairs separated by the lag ℓ ) obtained in simulations (Kritsuk et al. 2007) is similar to the velocity scaling observedin molecular clouds S ( δ u ) ∼ ℓ . (Heyer & Brunt 2004). Simulations also support theconcept of (lossy) energy cascade in compressible turbulence (e.g., V´azquez-Semadeni etal. 2003), suggesting that the kinetic energy directly lost in shocks constitutes a smallfraction of the total energy dissipation. The fact that the Richardson-Kolmogorov cascadepicture does approximately hold for supersonic turbulence follows from the linear scalingof the third-order structure function of the mass-weighted velocity, S ( δ √ ρ u ) ∼ ℓ , indi-cating constant turbulent energy transfer rate across the hierarchy of scales (Kritsuk etal. 2007; Kowal & Lazarian 2007; Schwarz et al. 2010). The power spectra of √ ρ u , accord-ingly, demonstrate the Kolmogorov scaling independent of the Mach number (Kritsuk etal. 2007; Schmidt et al. 2008; Kritsuk et al. 2009; Federrath et al. 2010; Price & Feder-rath 2010). It seems that this result can be also extended to supersonic MHD turbulence,where the incompressible 4 / S ±k , ≡ D δ Z ∓k ( ℓ )[ δ Z ± i ( ℓ )] E ∼ ℓ (here δ Z k ( ℓ ) ≡ [ Z ( x + ˆe ℓ ) − Z ( x )] · ˆe ,and ˆe ℓ is the displacement vector), if reformulated in terms of the mass-weighted Els¨asserfields Z ± ≡ ρ / ( u ± B / √ πρ ) (Kritsuk et al. 2009). The presence of magnetic field effec-tively reduces compressibility of the gas making the velocity spectra more shallow withslopes approaching the Iroshnikov-Kraichnan index of − . − . SM Turbulence i originally developed to study thermal, dynamic, and gravita-tional instabilities in shock-bounded slabs (Hunter et al. 1986; Vishniac 1994; Walder &Folini 1998; Folini et al. 2010). These models remain popular as a framework to directlysimulate star formation in molecular clouds (V´azquez-Semadeni et al. 2006; Hennebelle& Inutsuka 2006; V´azquez-Semadeni et al. 2007; Hennebelle et al. 2008; Inoue & Inutsuka2008-09; Heitsch et al. 2008-09; Banerjee et al. 2009; Niklaus et al. 2009; Audit & Hen-nebelle 2010; Rosas-Guevara et al. 2010). Recent numerical experiments with convergingflows have demonstrated strong sensitivity of results to adopted initial and boundary con-ditions as well as to model parameters that control the density of colliding gas streams,mean thermal pressure, orientation and strength of the mean magnetic field, levels andcharacter of “turbulence” at infinity, etc. All these parameters live their unique imprintsin the statistics of derived stellar populations and any comprehensive parameter studybased on computational modeling in this framework would be prohibitively expensive.One way to circumvent these difficulties is to exploit Prigogine’s concept of self-organization in nonequilibrium nonlinear dissipative systems (Nicolis & Prigogine 1977)in application to the ISM (e.g., Biglari & Diamond 1989). With this approach, one canuse interstellar turbulence as an agent that imposes “order” in the form of coherentstructures and correlations between various flow fields emerging in a simple periodic boxsimulation when a statistical steady state develops. In this case, the initial conditionsare no longer important, instead the steady state would provide the “correct” turbulentinitial conditions for star formation when self-gravity is turned on. While this idea is notnew, † it remained largely undeveloped so far. In the following sections we will discussthis concept in more detail and report first results from a series of MHD simulations ofturbulent multiphase ISM with the piecewise parabolic method on a local stencil (PPML;Ustyugov et al. 2009).
2. Self-organization in the magnetized multiphase ISM
In out numerical experiments, we treat the ISM as a turbulent, driven system, withkinetic energy being injected at the largest scales by supernova explosions, shear associ-ated with differential rotation of the galactic disk, gas accretion onto the disk, etc. (MacLow & Klessen 2004; Klessen & Hennebelle 2010). This kinetic energy is then beingtransferred from large to small scales in a cascade-like fashion. As our models include amean magnetic field, B , some part of this kinetic energy gets stored in the turbulentmagnetic field component, b , generated by stretching, twisting, and folding of magneticfield lines. The ISM is also exposed to the far-ultraviolet (FUV) background radiationdue to OB associations of quickly evolving massive stars that form in molecular clouds.This FUV radiation is the main source of energy input for the neutral gas phases and thisvolumetric thermal energy source is in turn balanced by radiative cooling (Wolfire et al.2003). The ISM is thus exposed to various energy fluxes, and self-organization arises as aresult of the relaxation through nonlinear interactions of different physical constituentsof the system subject to usual MHD constraints in the form of conservation laws. In thispicture, molecular clouds with their hierarchical internal structure form as dissipativestructures that represent active regions of highly intermittent turbulent cascade thatdrain the kinetic energy supplied by the driving forces. † See, for instance, summary of the panel discussion on Phases of the ISM during the 1986Grand Teton Symposium in Wyoming (Shull 1987).
22 Kritsuk, Ustyugov & Norman
Table 1.
Model parameters.
Model N n u rms , B β th , β turb , M A, cm − km/s µ GA 512 Figure 1.
Three snapshots of projected gas density in model A taken at t = 2, 3, and 4 Myr.The white-blue-yellow colors correspond to low-intermediate-high projected density values.
3. Modeling the formation of molecular clouds
To illustrate these ideas, we consider a set of simple periodic box models, which ignoregas stratification and differential rotation in the disk and employ an artificial large-scalesolenoidal force to mimic the supply of kinetic energy from various galactic sources. Thisnaturally leads to an upper bound on the box size, L , which determines our choice of L = 200 pc. Our models are, thus, fully defined by the following three parameters: themean gas density in the box, n ; the rms velocity, u rms , ; and the mean magnetic fieldstrength, B . All three would ultimately depend on L ; Table 1 provides the summary ofparameters for models A, B, C, D, and E assuming L = 200 pc. The table also gives thegrid resolution, N , the initial values for plasma beta, β th , ≡ πp /B , turbulent beta, β turb , ≡ πρ u , /B , and Alfv´enic Mach number, M A, = (4 πρ ) / u rms , /B , where p is the initial thermal pressure of the gas (see the phase diagram in Fig. 2 for moredetail).We initiate our numerical experiments with a uniform gas distribution in the compu-tational domain. An addition of small random isobaric density perturbations at t = 0triggers a phase transition in the thermally bi-stable gas that quickly turns ∼ − T = 184 K),while the rest of the mass is shared between the unstable and stable warm gas (WNM).In models A, B, and C, CNM and WNM each contain roughly ∼
50% of the total H i mass in agreement with observations (Heiles & Crutcher 2005). We then turn on theforcing and after a few large-eddy turn-over times the simulation approaches a statisticalsteady state. If we replace this two-stage initiation process with a one-stage procedureby turning the driving on at t = 0, the properties of the steady state remain unchanged.Figure 1 illustrates this evolutionary sequence for the two-stage case with three snap-shots of projected gas density for model A. The left panel shows two-phase medium at t = 2 Myr right before we turn on the forcing; the panel in the middle illustrates an earlystage of turbulization with transient “colliding flows” at t = 3 Myr. The right panel SM Turbulence B r m s , b r m s [ µ G ] Time [Myr]B rms , ABCDb rms , ABCD -2 -1 0 1 2 3log n [cm -3 ]2345 l og p t h / k B [ K c m - ] Model B , K , K K K Thermal Equilibrium
Figure 2.
Time evolution of the rms magnetic field strength, B rms , and its turbulent component, b rms , for models A, B, C, and D ( left panel). Phase diagram (thermal pressure versus gas density)for model B at t = 5 Myr ( right panel). -10-8-6-4-2 0 -2 -1 0 1 2 3 4 l og 〈 d N / N 〉 log n [cm -3 ] ABCn =5 cm -3 lognormal -4-3.5-3-2.5-2-1.5-1-0.5 2 2.5 3 3.5 4 4.5 5 l og d M / M log P gas /k B [cm -3 K] ABCDEHST
Figure 3.
Time-average density distributions for fully developed turbulence in models A, B,and C ( left panel) and time-average mass-weighted thermal pressure distributions for models A,B, C, and D ( right panel); see text for more detail. shows the projected density at t = 4 Myr for a statistically developed turbulent state.Molecular clouds can be seen in the right panel as filamentary brown-to-yellow structures(note that these are morphologically quite different from the transient dense structuresin the middle panel). The rms magnetic field is amplified by the forcing and saturateswhen the relaxation in the system results in a steady state, see Fig. 2. The level of satu-ration depends on B and on the rate of kinetic energy injection by the large-scale force,which is in turn determined by u rms and n . This level can be easily controlled with themodel parameters. In the saturated regime, models A and B tend to establish energyequipartition ( E K ∼ E M ), while the saturation level of magnetic energy in model C isa factor of ∼ p th , and density, n , separated by factors of 2.About 23% of the domain volume is filled with the stable warm phase at T >
T <
184 K) occupies ∼ ∼
70% of the volume residesin the thermally unstable regime at intermediate temperatures. The big orange dot atthe center indicates the (forgotten) initial conditions for models A, B, C, and E. Thephase diagram indicates that turbulence supports an enormously wide range of thermalpressures and also that p th in the molecular gas ( n >
100 cm − ) is higher than that inthe diffuse ISM, even though self-gravity is ignored in the model.24 Kritsuk, Ustyugov & Norman P th /k B [K cm -3 ]123456 l og P m ag / k B [ K c m - ] β t h = β t h = . P dyn /k B [K cm -3 ]0123456 l og P m ag / k B [ K c m - ] β t u r b = β t u r b = Figure 4.
Distributions of magnetic pressure vs. thermal ( left ) and dynamic ( right ) pressurefor a snapshot from model B at t = 5 Myr. -2 -1 0 1 2 3log n [cm -3 ]-2-1012 l og M A -2 -1 0 1 2 3 4log n [cm -3 ]-1.0-0.50.00.51.0 c o s ( θ ) Figure 5.
Distributions of the Alfv´enic Mach number and the cosine of the alignment anglevs. density for a snapshot from model A at t = 5 Myr. Figure 3 ( left ) shows the time-average density PDFs for models A, B, and C in thesteady state. The effect of magnetic field on the density PDF is apparently very weak onaverage, and the high-density part of the PDF can be well approximated by a lognormalas in the isothermal case. The signature of the two stable thermal phases in the PDF issmeared by the (relatively) high level of turbulence, u rms (100 pc) = 16 km/s (Brunt &Heyer 2004), but the overall shape of the distribution is not lognormal. The distributionof thermal pressure for models A, B, and C spans about 6 dex leaving no room forthe old pressure-supported cloud picture in the violent ISM. All distributions match thecharacteristic pressure typical for the Milky Way disk at the solar radius and show onlyweak dependence on B , while the width of the distribution remains sensitive to u rms and n . Figure 3 ( right ) shows how the mass-weighted pressure distributions obtained in ourmodels compare with the distribution reconstructed from high-resolution UV spectra ofhot stars in the HST archive (Jenkins & Tripp 2010). It seems that models with u rms =16 km/s reproduce both the shape and the width of the observed distribution quite nicely,while a lower turbulence level in model E makes that model distribution too narrow.These numerical experiments allow us to probe the levels of magnetic field strength inmolecular clouds that form self-consistently in the magnetized turbulent diffuse ISM. Inthe left panel of Fig. 4, we show a scatter plot of magnetic vs. thermal pressure for model Bat t = 5 Myr. The black contour lines show the distribution for the whole domain, which iscentered at β th ≈ .
1. The subset of cells representing the molecular gas (
T <
100 K and n >
100 cm − , color contour plot) shows very similar mean values of β th . This indicates atarget plasma beta for realistic isothermal molecular cloud turbulence simulations. The SM Turbulence right panel of the same Figure shows a scatter plot of magnetic vs. dynamic pressurefor the same snapshot from the same model. The distribution for the whole domain iscentered at β turb ≈ β turb ≈
30 meaning that turbulence in themolecular gas is super-Alfv´enic. Figure 5, left panel, shows the distribution of Alfv´enicMach number, M A , as a function of density for the strongly magnetized model A thatfurther supports this result. There is a clear positive correlation, M A ∼ n . , indicatedby the dashed line and most of the dense material ( n >
100 cm − ) clearly falls into thesuper-Alfv´enic part of the distribution.A key to understanding the origin of this super-Alfv´enic regime in the cold and densemolecular gas lies in the process of self-organization in magnetized ISM turbulence thatwe briefly introduced in Section 2. The statistical steady state that our models attainon a time-scale of a few million years is characterized by a certain degree of alignmentbetween the velocity and magnetic field lines. Figure 5, right panel, shows the distributionof the cosine of the alignment angle, cos θ ≡ B · u / ( Bu ), for model A at t = 5 Myr. Thecontours indicate a saddle-like structure of this probability distribution with a strongalignment regime (cos θ = ±
1) in the WNM at and around n ∼ − . This means thatcompressions in the WNM gas, which is on average trans-Alfv´enic (e.g., M A ∈ [0 . , . i , then turbulence in suchmolecular clouds can only be super-Alfv´enic, see also Padoan et al. (2010).
4. Conclusion
Rapid development of computational astrophysics in the recent years enabled progressin understanding the basics of interstellar turbulence. These new advances will help usto move forward with direct star formation simulations from turbulent initial conditions.
Acknowledgements . This research was supported in part by the National Science Foun-dation through grants AST-0607675, AST-0808184, and AST-0908740, as well as throughTeraGrid resources provided by NICS and SDSC (MCA07S014) and through DOE Officeof Science INCITE-2009 and DD-2010 awards allocated at NCCS (ast015/ast021).
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