The Janus Face of Turbulent Pressure
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1 CRC 963 Astrophysical Turbulence and Flow Instabilities with thanks to Christoph Federrath, Monash University Patrick Hennebelle, CEA/Saclay Alexei Kritsuk, UCSD yt-project.org Seminar über Astrophysik, Hamburg, November 2012
2 Overview Chandrasekhar (1951), etc.: Compressible turbulence increases the effective pressure of the gas: c 2 eff = c2 s u2 But is it really that simple?
3 Overview Chandrasekhar (1951), etc.: Compressible turbulence increases the effective pressure of the gas: c 2 eff = c2 s u2 But is it really that simple? Different routes: 1. Subgrid-scale turbulent pressure 2. Virial turbulent pressure 3.
4 Scale Separation in Turbulence Simulations
5 Decomposition of the Equations of Fluid Dynamics Compressible Navier-Stokes equation for momentum: ρu + (ρu u) = P + σ + F t Smoothing over length scale by a spatial filter yields: ρu + ρu u = P + σ +F t }{{} 0 if L/Re 3/4 Define ρ = ρ, ū = ρu / ρ, etc., then t ρū + ρu u = P + F Looks almost like the compressible Euler equation for ρū...
6 Turbulent Stresses... if we substitute the turbulence stress tensor τ = ρu u + ρū ū for the non-linear interactions across so that t ρū + ρū ū = P + τ + F Trace of τ defines the turbulent pressure: τ ii ρū 2 ρu 2 = 2ρK = 3P sgs ρk is the energy density of unresolved (l ) turbulent velocity fluctuations Effective pressure term in the Euler equation is given by P ( ( ) P + P sgs = P + 2 ) 3 ρk
7 How to Compute the Turbulent Pressure From the decomposition follows an additional PDE for ρk (e. g., Germano 1992, Schmidt et al. 2006): [ ] ρk + ( ρūk) = C t }{{} κ ρ K 1/2 K }{{} advection SGS transport (diffusion) + τ ij S ij 2 3 ρk d }{{} turbulent cascade C ɛ ρ K 3/2 }{{} dissipation Production rate Σ = τ ij Sij, where (Woodward et al. 2006, Schmidt & Federrath 2011) τ ij = C 1 ρ K 1/2 S ij }{{} linear eddy-viscosity part 2C 2 ρk 2ū i,kū j,k ū 2 } {{ } non-linear part Supersonic turbulence: C and C (1 C 2) ρkδ ij
8 The subgrid-scale turbulent pressure LES of Supersonic Turbulence Denstrophy 12 (ρ1/2 v) 2 in a 5123 LES (Enzo), Mrms 5.5 (Schmidt & Federrath 2011) solenoidal forcing compressive forcing
9 The subgrid-scale turbulent pressure LES of Supersonic Turbulence SGS turbulence energy ρk in a 5123 LES (Enzo), Mrms 5.5 (Schmidt & Federrath 2011) solenoidal forcing compressive forcing
10 SGS Turbulence Energy Statistics K sgs Ρ 0 V LES with grid resolutions changing from 64 3 to : Mean SGS turbulence energy follows power-laws with slope β 0.8 (steeper than Kolmogorov scaling) K sgs Ρ 0 V 2 K sgs Ρ 0 V
11 Effective Pressure Microscopic equation of state for isothermal gas: P ρ Effective pressure on the grid scale: P eff = P+ 2 3 ρk ρc2 0 ( ) 3 M2 sgs LES, solenoidal forcing, M rms Locally, the turbulent pressure can largely exceed the thermal pressure 1 cdf sgs log 10 P eff P log 10 Ρ Ρ 0
12 The General Virial Theorem The Euler equation including gravitational forces can be contracted with r and integrated over the volume V By defining I = ρr 2 d 3 x, E int = V E int = V φ E grav = ρx i d 3 x, E kin = V E kin = x i V it follows that 1 2Ï = 2(E int + E kin ) + E grav + V V V 3 2 P d3 x 1 2 ρu2 d 3 x ( Pδij + u 2) x i n j ds
13 Virial Equilibrium Under the assumptions equlibrium (Ï = 0) surfaces terms negligible no external gravitational potential we obtain the well-known relation 2(E int + E kin ) + E grav 0 The first two terms can be expressed as where 2(E int + E kin ) = 3 (P + P t ) P = P d 3 x and P t = 1 V 3 V ρu 2 d 3 x Applicable to molecular clouds, even to substructure in molecular clouds (clumps, cores)?
14 Numerical Clump Mass Distributions Schmidt, Kern, Federrath, and Klessen (2010): Isothermal turbulence simulations with assumed mass scale given by L = 0.6 pc and n = 10 4 cm 3 (λ 0 J /L 0.2) Simply connected regions exceeding their Bonnor-Ebert mass are identified by a clump finding algorithm (Padoan & Nordlund 2007) 2E int = 3P E grav: 3(P + P t) E grav:
15 Hennebelle-Chabrier Theory of the Clump Mass Function Hennebelle & Chabrier (2008): Based on the Press-Schechter statistical formalism for hierarchical objects, the mass distribution is related to the PDF P(δ) of δ = log(ρ/ ρ): N (M) ρ ( M dm dr ) 1 dδ dr exp(δ)p(δ) Clump scale R is connected to its mass M By assuming the Jeans- or Bonnor-Ebert-mass for gravitationally unstable cores, we have M = R = exp( δ/2), Can now calculate N (M) for given P(δ) if R L where M = M R MJ 0, R = λ 0 J
16 The Clump Mass Function with Turbulent Support Define effective Jeans mass M J,eff through the effective sound speed (in virial equilbr.) ceff 2 = c δu2 (R) [ c ( ) η ] R 3 M rms L New parameter: turbulent Mach number scaled down to Jeans length at the mean density 3 M = M rms ( λ 0 J /L ) η Can compute N (M) for given P(δ), M, and η for HD turbulence with soln./compr. forcing (Federrath et al. 2010, Schmidt et al. 2010) With these definitions: ( ( M = R 1 + M 2 R 2η) 1 + M 2, δ = log R 2 R 2η )
17 Turbulent Clouds Isolated system: virial equilibrium, turbulent pressure calculated by Bonazolla (1992) Substructure: boundary terms, turbulent pressure? white
18 Caveats Bonazzola et al. (1992) showed that effective sound speed is only valid on( length scales integral length scale CMF theory and numerical analysis are based on the assumption of local virial equilibrium for the critical mass Questioned by Ballesteros-Paredes (2006) Dib et al. (2007) demonstrate with numerical data that cores/clumps are non-equilibrium structures: significant contributions from boundary terms undergoing supersonic compression while not necessarily gravitationally bound Flat slope can also be explained by magnetohydrodynamic turbulence without turbulent pressure support
19 The Rate of Compression Equation for the rate of compression (d = u): Dd Dt = 2 φ Λ turb Λ therm = 4πG(ρ ρ 0 ) }{{} gravity 1 ( ω 2 S 2) 1 ( 2 P + 1ρ ) } 2 {{} ρ ρ P }{{} turbulence thermal pressure hydrostatic equilibrium for d = ω = S = 0 Jeans criterion follows in the limit δ = ρ/ρ free-fall collapse with τff (Gρ) 1/2 for 4πG(ρ ρ 0 )/ Λ 0 Schmidt, Collins, and Kritsuk (2012): Compute statistics of positive/negative support for self-gravitating turbulence simulations: Λ + = { Λ if Λ 0 0 otherwise Λ = { Λ if Λ 0 0 otherwise
20 Self-Gravitating Turbulence Kritsuk et al. (2011): α 0.25 and M rms 6 Deep AMR simulation with 5 levels of refinement, refinement factor 4
21 Thermal Support Mean thermal support is positive Asymptotic relation: 2 ρλ therm P for P 100 Intermittency: strong fluctuations produced by filaments/cores undergoing strong compressions Strong support around δ 10 7 corresponds to flattening of the density PDF
22 Turbulent Support However, mean turbulent support is negative Asymptotic relation: 2 ρλ turb ρω 2 for ρω 2 1 Negative pressure caused by shear and supersonic compression overcompensates positive pressure of turbulent eddies
23 Power Spectra 500 P Ω k P turb k P S k k 2 Fourier-transform of root-grid data (smoothing factor up to 1024) Burgers scaling ω(l) l 1/2 implies a flat power spectrum: ( ) 1 P turb (k) = 2 ˆω i ˆω i ŜijŜij k 2 dω k = 1 2 [P ω(k) P S (k)] k =k
24 Total Support For 100 δ 10 4, net support has negative floor with Λ 4πGρ 0 δ: supersonic gas compression triggers gravitational collapse this corresponds to the tail of the density PDF with slope to the 1.7 (compared to 1.5 for pressure-free collapse solution by Penston 1969)
25 Resume In the LES picture, something like Chandrasekhar s effective pressure follows rigorously from scale separation Applies approximately to substructure (clumps/cores) in turbulent star-forming clouds if local virial equilibrium can be assumed and boundary terms can be neglected Turbulent support against gravity appears to be negative in the non-linear non-equilibrium regime, so the nature of turbulent pressure is profoundly different from the thermal and magnetic pressures Are these conclusions somehow in contradiction? No, because Turbulent pressure is just the diagonal part of the full turbulence stress tensor Support is locally given by the pressure Laplacian, not the pressure itself
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