Subgrid Scale Physics in Galaxy Simulations

Transcription

1 CRC 963 Astrophysical Turbulence and Flow Instabilities with thanks to Harald Braun, Jan Frederic Engels, Jens Niemeyer, IAG Ann Almgren and John Bell, LBNL Christoph Federrath, Monash University yt-project.org Galactic Scale Star Formation, Heidelberg, July/August 2012

2 Objective There are many star formation and feedback recipes for simulations (Robertson & Kravtsov 2008, Agertz et al. 2009, Tasker & Tan 2009, Bournaud et al. 2010, Dobbs & Pringle 2010, Governato et al. 2010, Greif et al. 2010, etc.) We do not aim at galaxy simulations in a static environment with resolution 1 pc: want to study galaxies in their fully dynamical cosmological environment, including galactic outflows apply a subgrid scale model for the multiphase turbulent ISM (Braun & WS 2012) simulations of isolated disk galaxies mainly serve as as a testing case for the model

3 A Simple Two-Phase Model Split mass contents of grid cells into cold and warm phases with average densities ρ c,pa = m c /V c and ρ w,pa = m w /V w (Springel & Hernquist 03):

4 Effective Pressure Equilibrium Basic assumption: two-phase structure given by generalized virial theorem for ensemble of cold-gas clouds embedded in the warm medium: 3P c,eff π }{{} 10 Gρ2 c,palc 2 3P w,eff 0 }{{}}{{} int. + kin. grav. ext. Effective pressure P c,eff = ρ c,pa σc,eff 2 (Chandrasekhar 1951): ( σc,eff 2 = c2 c 1 γ + σ2 c,turb = γ(γ 1)e c γ + 1 ) 3 M2 c,turb If the bulk of the cold gas is not strongly self-gravitating, then P c,eff P w,eff implies ρ c,pa ρ w,pa = σ2 w,eff σ 2 c,eff

5 Star Formation Model Cold gas is converted into star particles at a rate ( SFR ff f H2 ρ c 3π ρ s = ɛ core, where t c,ff = t c,ff 32Gρ c,pa ) 1/2 Molecular gas fraction f H2 = m H2 /(ρ c 3 ) is determined by a Strömgren-like approach similar to Krumholz et al Dimensionless star formation rate per free fall time is given by (Padoan & Nordlund 2011) SFR ff = x crit xp(x) dx, where x crit σ2 c,turb πgρ c,pa lc }{{ 2 } α vir M 2 c,turb Turbulent density PDF p(x) is assumed to be log-normal with variance (Federrath et al. 2010) σ 2 ln ( 1 + b 2 M 2 c,turb), where b = 1/3 (soln.) or 1 (compr.)

6 Composite optical HST and Chandra X-ray image of supernova 1987a

7 Supernova Feedback Model Supernova rate is determined by the star formation rate and the Chabrier (2001) fit to the IMF: ρ s,fb (t) = te t b ρ s (t t ) IMF(m ) dm dt dt, Increase of warm-gas thermal energy due to heating and cold-gas evaporation (McKee & Ostriker 1977): d(ρ w e w ) dt = [(1 ɛ SN )e SN +Ae c ] ρ s,fb, where e SN erg/m SN Production of turbulent pressure P turb = 2 3 ρk: d(ρk) dt = ɛ SN e SN ρ s,fb, where ɛ SN SN

8 The Euler Equations with Subgrid-Scale Dynamics Couple Euler equations for resolved flow variables to unresolved turbulence energy ρk such that ρ(e + K) is conserved: t t ρ + (uρ) = 0 (ρu) + (ρu u) = (P+ 23 ) ρk }{{} ρe + (ρue) = t eff. pressure [ u + τ sgs }{{} +ρ(g + f ext ) nondiag. stresses (P+ 23 )] ρk + (u τ sgs) + ρu (g + f ext ) Λ + Γ }{{} radiative Σ + ρɛ }{{} turbulent ρk + (ρuk) = D + Σ ρɛ (Schmidt et al. 2006) t

9 Closures for the Compressible Turbulent Cascade Production rate of SGS turbulence energy Σ = τ ij S ij, where (Woodward et al. 2006, WS & Federrath 2011) τ ij = C 1 ρk 1/2 S ij }{{} linear eddy-viscosity part 2C 2 ρk 2u i,ku j,k u 2 } {{ } non-linear part C 1, C 2 : turbulent pressure given by 2 3 (1 C 2)ρKδ ij P turb = 2 3 ρk = 1 3 τ ii Closure coefficients C and C depend only little on the Mach number in the supersonic regime Turbulent dissipation rate ɛ = C ɛ K 3/2 /

10 LES of Supersonic Turbulence on Nested Grids Vorticity modulus ω SGS turbulence energy ρk Forced turbulence in a periodic box (128 3 root grid) Cooling L = χρ(e e 0 ) keeps int. energy roughly constant Statistically stationary state after 2 integral time scales

11 LES of Supersonic Turbulence on Nested Grids M rms 5 Mean SGS turbulence energy rms sgs ΡK t Turbulent Mach numbers: ratio of resolved/unresolved kinetic to thermal energies SGS turbulence energy scales down with refinement level Variables from higher levels are averaged down to coarser cells and energy correction is applied! t

12 The Role of Subgrid Scale Turbulence M c,turb = 3σ c,turb /c c, where 3σ 2 c,turb = 2K(λ J,c/ ) 2η

13 SGS Turbulence Energy Equation for Galaxy Simulations t ρk + (ρuk) = D + ɛ SNe SN ρ s,fb + (1 f th )ɛ tt Λ eff ρ w }{{} internal + (τ ij ) sgs S ij } {{ } external 2 3 ρkd ρc K 3/2 ɛ Internal driving: Production by supernova feedback e SN ρ s,fb Production by thermal instability ρ w Λ eff, where Λ eff = Λ rad Γ PAH Γ Lyc ɛ is the effective cooling rate External driving: Production through turbulent cascade from length sales L l Coupling to resolved turbulence driven by gravity and shear of the disk

14 The Cosmological Fluid Dynamics Code Nyx Initiators: Jens Niemeyer (IAG), Peter Nugent (LBNL) Code paper: Almgren et al. (ApJ submitted) Boxlib framework for block-structured AMR Hybrid OMP/MPI parallelization for up to cores Unsplit-PPM hydro solver Multi-grid Poisson solver for self-gravity PM treatment of dark matter/star particles CLOUDY cooling SGS model for turbulent multiphase ISM

15 Initialization: Stable Adiabatic disk Inititalization: Collapse into a thin disk after 100 Myr: IC: stable rotating disk with (r, θ)-profile of Wang et al and initial temperature K Static DM halo, M baryons (Z/Z = 0.1) in a 1 Mpc 3 box Development runs: root grid, 8 refinement levels (30 pc resolution)

16 Evolution of Star Formation and Feedback

17 Star Formation (300 Myr) Local star formation rate ρ s vs. ρ Local star formation rate ρ s vs. ρ H2

18 Cold gas and Turbulence (300 Myr) Fraction of cold gas ρ c /ρ vs. ρ Typical star formation efficiency ɛ ff 0.01 for 1 < M sgs < 10

19 Warm-Gas Fraction and Stars (300 Myr) Fraction of warm gas ρ w /ρ Stellar mass in M /pc 3

20 Conclusions Cooling and gravity initially form large massive clumps in a gas-dominated adiabatically stable disk cannot be prevented by feedback, although feedback subsequently strips off gas clumps may merge and migrate toward the center to form a bulge or be torn apart and further fragment Turbulence-regulated star formation saturates quickly at a few solar masses per year Thermal and turbulent feedback pressurizes disk and drives galactic outflows