mhd simulations of interstellar turbulenceakpc.ucsd.edu/multiphase/losalamos2010.pdf · mhd...
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MHD Simulations of Interstellar Turbulence 1
Alexei KritsukUniversity of California, San Diego
Collaborators: Mike Norman (SDSC), Sergey Ustyugov (Keldysh) & Rick Wagner (SD SC)
http://akpc.ucsd.edu LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Outline
Grand Challenges: Model star formation ab initio, explain the origin of the
IMF, predict SFR in different environments
Interstellar turbulence is the key organizing process in the ISM
Major stumbling blocks:
• compressibility
• magnetism
• self-gravity
• numerics
Scaling, intermittency & compressible cascade
Self-organization in compressible MHD turbulence
A role for self-gravity in supersonic turbulence
Summary, Perspective
MIST: Magnetized Inter stellar Turbulence LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Dust structures within 150 parsecs of the Sun
3
IRAS, 100 micron
Planck HFI (55x55 deg., 540 & 350 micron)
Hershel PACS/SPIRE, the Eagle, ~20 pc across
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Large-scale structure of the molecular gas in Taurus
4
The 12CO column density (cm−2) distribution; 21×26 pc [Goldsmith et al., 2008]
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Universal linewidth-size relation in MCs
5
Interstellar turbulence within MCs is invariant over a wide range of scales
Sound speed (T=10K)
Reynolds number: Re = u(L)Lν
∼ 108
Outer scale: L & 50 pc
Mach number: Ms(L) ≡ urms
cs≫ 1
Velocity scaling: S1(ℓ) ∼ ℓ0.56±0.02
Filled circles – global velocity dispersion
and size for each cloud.
Heavy solid line is equivalent to Larson’s
(1981) relation.
The composite relation from PCA decompositions of 12CO J=1-0 imaging observations of 27
molecular clouds, δu = (0.87±0.02)ℓ0.65±0.01, corresponds to a 1st-order structure function
scaling: S1(ℓ) ∼ ℓ0.56±0.02 [Heyer & Brunt, 2003-04].
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Mass–radius relation for a sample of 580 MCs
6
The fractal dimension of the ISM is around 2.36
Based on UMSB survey [Roman-Duval, Jackson, Heyer, Rathborne, & Simon, 2010].
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Magnetic field direction in Taurus
7
The 13CO antenna temperature distribution integrated over v ∈ [5,8] km/s
The line segments indicate the magnetic field direction derived from absorption by polarized
dust grains [Heiles, 2000; Goldsmith et al., 2008]
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Young stars and molecular gas in Taurus
8
Goldsmith et al. (2007)
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Outline
9
I. Scaling, intermittency,and compressible cascade
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
ENZO simulation, 2008
10
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Simulations reproduce the fractal structure of MCs
11
The mass dimension, Dm, is 2 on small scales (shock fronts) and≈ 2.3 in the inertial range; compare to Dm = 2.36±0.04 derived
from observations [Roman-Duval et al., 2010]
2.6
2.7
2.8
2.9
3
3.1
0.5 1 1.5 2 2.5 3
log 1
0 M
(ℓ)/
ℓ
log10 ℓ/∆
Dm=2.28±0.01Dm=2
Mach 6
Isothermal gas dynamics at 20483, Ms = 6 [Kritsuk et al. 2009]
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Simulations reproduce the observed scaling
12
First-order velocity structure functions:S1(uuu,ℓ) ≡ ⟨|uuu(rrr +ℓℓℓ)−uuu(rrr )|⟩ ∼ ℓ0.54±0.01
-0.4
-0.2
0
0.2
0.4
0.6
0.5 1 1.5 2 2.5
log 1
0 S
(ℓ)
log10 ℓ/∆
0.533(2)0.550(4)
LongitudinalTransverse
Isothermal gas dynamics at 20483, Ms = 6 [Kritsuk et al. 2006-09]
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
First von Kárma n–Howarth relation ≈ holds
13
Second-order velocity structure functions:S2(uuu,ℓ) ≡ ⟨|uuu(rrr +ℓℓℓ)−uuu(rrr )|2⟩ ∼ ℓ0.96±0.01
-0.5
0
0.5
1
1.5
0.5 1 1.5 2 2.5
log 1
0 S
(ℓ)
log10 ℓ/∆
0.952(4)0.977(8)
LongitudinalTransverse
S⊥2 = 4
3S∥
2 converts into S⊥2 ≈ 1.27S∥
2 at Ms = 6 [Kritsuk et al. 2007]
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
The 4/5-Law breaks
14
Third-order velocity structure functions:S3(uuu,ℓ) ≡ ⟨|uuu(rrr +ℓℓℓ)−uuu(rrr )|3⟩ ∼ ℓ1.27±0.02
-0.5
0
0.5
1
1.5
2
2.5
0.5 1 1.5 2 2.5
log 1
0 S
(ℓ)
log10 ℓ/∆
1.26(1)1.29(1)
LongitudinalTransverse
S∥3(uuu,ℓ) does not scale linearly with ℓ at Ms = 6 [Kritsuk et al. 2007]
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Compressible Navier-Stokes flux relation holds
15
Exact current-density correlation function [Falkovich et al. 2010]Φ(ri ) ≡∑
j ⟨ρ(0)ui (0)[
ρ(rrr )u j (rrr )ui (rrr )+p(rrr )δi j
]
⟩ = ǫri
3
0
1
2
3
4
5
6
7
8
9
0 20 40 60 80 100 120 140
Φ(r
)
r/∆
10243 Mach 6Linear Fit
Φ(r ) does scale linearly with r at Ms = 6 [Wagner et al. 2011, in prep.]
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Compressible cascade à la Richardson-Kolmogorov
16
• Simple dimensional arguments:
Energy cascade in incompressible turbulence:
(δu)2(
δuℓ
)
≡ const ⇒ (δu)3 ∼ ℓ⇒ (δu)p ∼ ℓp3 [Kolmogorov 1941]
Energy cascade in supersonic turbulence:
ρ(δu)2(
δuℓ
)
≡ const [e.g., Lighthill 1955] ⇒ ρ(δu)3 ∼ ℓ
δv ≡ ρ13 δu ⇒ (δv)p ∼ ℓ
p3
These scaling laws (both K41 and compressible) do not include intermittency corrections.
Using the density-weighted velocity v instead of u, one properly accounts for density–velocity
correlations in compressible flows.
[Henricksen (1991); Fleck (1996); Kritsuk et al. (2007); . . . ]
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Intermittency in supersonic turbulence
17
0
0.5
1
1.5
2
0 1 2 3 4 5 6
ζ p
p
Absolute scaling exponents for SFs of u and v=ρ1/3u
K41SL94BurgB02
M6 T uM6 T v
Kritsuk et al. (2006-10); Kowal & Lazarian (2007-10); Schwarz et al. (2010); Price & Federrath
(2010); Falkovich et al. (2010); Schmidt et al. (2008-09); Federrath et al. (2010)
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Summary: the 1/3-rule
18
Regime Incompressible Highly CompressibleMs ≪ 1 Ms > 3
4/5-Law 4/5-Law + 1/3-rule
HD [Kolmogorov 1941] [Kritsuk et al. 2007]
⟨[δu∥(ℓ)]3⟩ =− 45ǫℓ ⟨|δv(ℓ)|3⟩∝ ℓ
MA =∞ Velocity field: u Modified velocity: v ≡ ρ1/3u
4/3-Law 4/3-Law + 1/3-rule ???
MHD [Politano & Pouquet 1998] [Kritsuk et al. 2009]
⟨[δz∓∥ (ℓ)[δz±
i(ℓ)]2⟩ =− 4
3ǫ±ℓ ⟨[δZ∓
∥ (ℓ)[δZ±i
(ℓ)]2⟩∝ ℓ
MA =O(1) Elsässer fields: z± ≡ u±B Elsässer′: Z
± ≡ ρ1/3(u± Bpρ
)
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Outline
19
II. Effects of magnetic field and EOS
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Multiphase ISM as a dissipative system
20
À la Prigogine:Self-organization governed by the energy flux through the nonlinear system
Thermal pressure , pth ≡ (γ−1)ρe; Heating and cooling, Γ−nΛ(T )
• Thermal beta: βth ≡ pth/pmag = (γ−1)ETh/EM
Magnetic pressure , pmag ≡ B 2/8π; Uniform field B0
• Turbulent beta: βturb ≡ pdyn/pmag = 2M 2A = 2EK/EM
• Alfvén Mach number: MA ≡ u/v A , Alfvén speed: v A ≡√
2pmag/ρ
Dynamic pressure , pdyn ≡ ρu2; Isotropic solenoidal forcing F
• Ratio: βturb/βth = pdyn/pth = γM 2s = 2/(γ−1)EK/ETh
• Sonic Mach number: Ms ≡ u/cs , sound speed: cs ≡√
γpth/ρ
Gravity , TBD
Developed turbulence does not depend on the way it was produced
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Basic equations
21
• Ideal compressible MHD with a volumetic energy source (mass, momentum, flux, energy):
∂ρ
∂t+∇· (ρu) = 0, (1)
∂ρu
∂t+∇·
[
ρuu−BB+(
p + B2
2
)
I
]
= F, (2)
∂B
∂t+∇· (uB−Bu) = 0. (3)
∂E
∂t+∇·
[(
E +p + B2
2
)
u− (B ·u)B
]
= u ·F+ρΓ−ρ2Λ(T ). (4)
• Pressure: p ≡ (γ−1)eρ, γ= 5/3; Specific internal energy density e
• Total energy density: E = ρe +ρu2/2+B 2/2
• Volumetric heating and cooling rates: Γ, ρΛ(T )
• Solenoidal constraint on B: ∇·B ≡ 0
• Forcing: F ≡ ρa−⟨ρa⟩, Fixed large-scale solenoidal non-helical acceleration: a(x)
• Initial conditions: ρ0 +δρ, p0, u0 = τa, B0 = (B0,0,0); Periodic boundary conditions
• Implicit large eddy simulation (ILES) approach [e.g., Sytine et al. (2000)]
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Observational constraints
22
ISM thermodynamics [Wolfire et al., 2003] PDF of thermal pressure [Jenkins, 2010]
-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5log
10 n [cm-3]
3.0
3.2
3.4
3.6
3.8
4.0lo
g 10 p
th/k
B [K
cm
-3]
A, B, C, E
D
Models
10,0
00 K
5,25
0 K
184
K
18 K
Thermal Equilibriumstable unstable -4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
2 2.5 3 3.5 4 4.5 5
log 1
0 dN
/N
log10 Pth/kB [cm-3K]
Mass-weighted PDF of Thermal Pressure
Linewidth-size relation [Brunt & Heyer, 2004] B–n relation [Crutcher et al., 2010]
Models A, B, C, D
E
Sound speed
0 2 4 6 8log
10 n(H) [cm-3]
-2
0
2
4
log 10
I<B
los>
I [µG
]
HI OH CN
Crutcher et al. (2010)
Falgarone et al. (2008)
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Time-evolution of density structures
23
Projected gas density at t = 20 Myr (left) and at t = 27 Myr (right) for Model A
← Two-phase medium right after the forcing is turned ONTransient “colliding flows” initialize multiphase turbulence →
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Time-evolution of density structures
24
Projected gas density at t = 39 Myr (left) and at t = 46 Myr (right) for Model A
Two snapshots illustrate the statistically dveloped turbulent stateNB: Structures appear very different from those seen as transients
Movie
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
High-density structures
25
Full box (left) and a 25% slab (right) from Model A
Dense material is assembled in hierarchical filamentary structuresLarge molecular complexes contain comparable amounts of atomic gas
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Magnetic structures
26
Current sheets (left) and regions of high B 2 (blue, right) in Model A
Magnetic dissipative structures simulated with PPML are very sharp
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Global energetics
27
Kinetic, magnetic, and thermal energy density versus time
-5
-4
-3
-2
-1
0
1
0 10 20 30 40 50 60
log 1
0 E
Time [Myr]
EK, Model ABC
EM, Model ABC
ETh, Model ABC
0
2
4
6
8
10
12
14
16
18
0 10 20 30 40 50 60
Brm
s, b
rms
[µG
]
Time [Myr]
Brms, ABCD
brms, ABCD
• Models A, B & C have uniform magnetic fields B0 = 10, 3, & 1 µG, respectively
• Model A demonstrates global energy equipartition (EK ∼ EM )
• Models B and C stop short of reaching the equipartition
• Model C: brms continues to grow linearly after 6tdyn; stationary regime emerges later
• Dynamical time tdyn ≡ L/2urms = 6.1 Myr
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Thermal pressure versus density
28
Turbulence supports a wide range of thermal pressures;pth in molecular gas is higher than that in the diffuse ISM
-2 -1 0 1 2 3log
10 n [cm-3]
2
3
4
5lo
g 10 p
th/k
B [K
cm
-3]
Model B
10,00
0 K
5,250
K
184 K
18 K
Thermal Equilibrium
• Heavy line indicates thermal equilibrium: nΛ(T ) = Γ
• Orange circle shows the initial conditions for Model B
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Density distribution
29
Time-average gas density distribution for models A, B & C
-10
-8
-6
-4
-2
0
-2 -1 0 1 2 3 4
log 1
0 ⟨d
N/N
⟩
log10 n [cm-3]
ABC
n0=5 cm-3
• Not a log-normal; contains a signature of the two thermal phases; peaks at 1 cm−3 < n0
• Weak dependence on B0, in particular, at large n
• Statistical samples include 9.4×109 cells per snapshot (70 flow snapshots at 5123)
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Density distribution
30
Density-weighted distribution: models vs. observations
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
2 2.5 3 3.5 4 4.5 5
log 1
0 dM
/M
log10 Pgas/kB [cm-3K]
ABCDE
HST
• Lines represent time-average distributions for lines of sight with N (HI) < 2.5×1021 cm−2
• Data points from high-resolution UV spectra of hot stars in the HST archive [Jenkins, 2010]
• Model E is rejected: urms,0(E) is too small; Model D is marginal: n0(D) is small
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
B-n diagram
31
-2 0 2 4 6 8log
10 n [cm-3]
-2
0
2
4
log 10
IBI/2
, log
10I<
Blo
s>I [
µG]
HI OH CN
• Isocontours represent data snapshot from Model B
• Observational data points: • Crutcher et al. (2010), • Falgarone et al. (2008)
• Model B matches the HI Zeeman data from Crutcher et al. (2010)
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Magnetic versus dynamic pressure
32
With sufficient magnetization, the system finds a balancepdyn ∼ pmag ⇒ βturb ∼ 1
0 2 4 6 8log
10 P
dyn/k
B [K cm-3]
0
1
2
3
4
5
6
log 10
Pm
ag/k
B [K
cm
-3]
βturb=
1
• Isocontours represent data snapshot from Model B (log spacing)
• Dash-dotted line indicates βturb = 1
Where is the molecular gas on this diagram?MIST LANL Astro Seminar – December 20, 2010
Alexei Kritsuk
Magnetic versus dynamic pressure
33
Molecular clouds are born super-Alfvénic with βturb > 30
0 2 4 6 8log
10 P
dyn/k
B [K cm-3]
0
1
2
3
4
5
6
log 10
Pm
ag/k
B [K
cm
-3]
βturb=
1
βturb=
30
• Isolevels for a subset of cells with the cold (T < 100 K) and dense (n > 100 cm−3) material
representative of the molecular gas are shown in color
• Black isocontours are the same as on previous slide
• Dashed line indicates βturb = 30, dash-dotted: βturb = 1
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Magnetic versus thermal pressure
34
Plasma beta in molecular clouds: βth ≈ 0.1Now we can calibrate βth,0 in our isothermal models of MC turbulence!
1 2 3 4 5log
10 P
th/k
B [K cm-3]
1
2
3
4
5
6
log 10
Pm
ag/k
B [K
cm
-3]
βth=1
βth=0.1
• Black isocontours represent data snapshot from Model B
• Isolevels for a subset of cells with the cold (T < 100 K) and dense (n > 100 cm−3) material
representative of the molecular gas are shown in color
• Dashed line indicates βth = 0.1, dash-dotted: βth = 1
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Alfvén Mach number versus density
35
Even in Model A, dense gas (n > 100 cm−3) is super-Alfvénic
-2 -1 0 1 2 3log
10 n [cm-3]
-2
-1
0
1
2
log 10
MA
• Isocontours represent data snapshot from Model A
• Dashed line indicates the best fit scaling, MA ∼ n0.4
• Horizontal line separates the sub- and super-Alfvénic regions
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
PDF of the alignment angle
36
Distribution of cosθ ≡ B·uBu
shows strong alignment of B and u at large B0
0
1
2
3
4
5
6
7
-1 -0.5 0 0.5 1
log 1
0 ⟨d
N/N
⟩
cos θ
ABC
-2 -1 0 1 2 3 4log
10 n [cm-3]
-1.0
-0.5
0.0
0.5
1.0
cos(
θ)• B−u alignment is most pronounced in Model A where EK ∼ EM
• Alignment is strong in the bulk of the volume (trans-Alfvénic turbulence)
• Alignment is weak at low densities and at high densities
• Model C shows no significant alignment because EK ≫ EM
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Dilatational-to-solenoidal ratio
37
Strong magnetization effectively reduces compressibility
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
0 0.5 1 1.5 2 2.5
log 1
0 χ(
k)
log10 k/kmin
ABC
• χ(k) ≡ P (ud,k)/P (us,k); Helmholtz decomposition: u = us +ud
• 3D compressions are suppressed in strongly magnetized models
• Large-scale compressions from forcing are missing in Model A
• Purely solenoidal forcing is OK for Model A [Kritsuk et. al, 2010]
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
PDF of turbulent magnetic field components
38
Distribution of b⊥ is platykurtic for Model A and leptokurtic for models B & CDistribution of b∥ is negatively skewed for large positive B0
-5
-4
-3
-2
-1
0
-40 -30 -20 -10 0 10 20 30 40
log 1
0 ⟨d
N/N
⟩
b⊥ [µG]
B0(A)B0(B)B0(C)
ABC
-5
-4
-3
-2
-1
0
-50 -40 -30 -20 -10 0 10 20 30 40
log 1
0 ⟨d
N/N
⟩
b‖ [µG]
B0(A)B0(B)B0(C)
ABC
• Vertical dashed lines show B0 = 9.5, 3.0, and 0.95 µG (models A, B, and C, respectively)
• Skewness γ1 ≡µ3/σ3 =−1.0, −0.2, and 0.2 for models A, B, and C
• Excess kurtosis γ2 ≡µ4/σ4 −3 =−0.4, 0.3, and 3.6 for models A, B, and C
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
PDF of the magnetic field strength
39
B0 controls the abundance of cold dense gas with extreme field values
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
0 20 40 60 80 100 120 140
log 1
0 ⟨d
N/N
⟩
B [µG]
B0(A)=10µGB0(B)=3µGB0(C)=1µG
ABC
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
0 20 40 60 80 100 120
log 1
0 ⟨d
N/N
⟩
B [µG]
Full distributionn < 2 cm-3
2 < n < 30 cm-3
n > 30 cm-3
• Chances to find B ∈ [100,120] µG in weakly magnetized Model C are eight times higher
than in strongly magnetized Model A (magnetic tension)
• Dense & cold material dominates the high end of the distribution
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Summary: magnetism
40
Many diagnostics are only weakly sensitive to magnetic effects
Magnetic and dynamic pressure dominate in the ISM, thermal effects are
subdominant
While there is a tendency to global energy equipartition (EM ∼ EK ), no
detailed scale-by-scale energy balance exists
Kinetic energy always dominates on small scales; molecular clouds are
born super-Alfvénic , while the WNM/WIM can be sub-Alfvénic
In models with large B0, strong magnetic tension suppresses 3D
compressions and reduces the level of magnetic fluctuations
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Outline
41
III. Effects of self-gravity
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
A deep AMR simulation: 5123L5×4 (2004)
42
5 pc
0.02 pc
300 AU
Kritsuk, Wagner, & Norman (2011)
• To correctly reproduce statistics of self-gravitating prestellar cores, one needs to model the
hierarchical structure of turbulent MCs on scales from 10 pc to 0.01 pc.
• AMR helps to follow the evolution of gravitationally unstable objects from turbulent initial
conditions, their fragmentation and collapse.
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Simulated filamentary stucture of molecular clouds
43
.
Rendering : NCSA Advanced Scientific Visualization Laboratory: Donna Cox, Matt Hall, AJ Christensen,
Bob Patterson, Stuart Levy
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Formation of protostellar cores in molecular clouds
44
.
Rendering : NCSA Advanced Scientific Visualization Laboratory: Donna Cox, Matt Hall, AJ Christensen,
Bob Patterson, Stuart Levy (animation)
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Column density PDFs: Star-forming MCs
45
Wide-field dust extinction map of the Taurus MC complex18×18 pc; N (H2+ H)/AV = 9.4×1020 cm−2 [Bohlin et al., 1978]
• Left: linear AV ; Right: logarithmic AV [Kainulainen et al. 2009]
• The contour at AV = 4 mag shows where the column density PDF deviates from lognormal
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Column density PDFs: Star-forming MCs
46
Kainulainen et al. (2009)
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Column density PDFs: Non-star-forming MCs
47
Kainulainen et al. (2009)
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Density PDF with deep AMR
48
Extended power law tail: > 6 dex in density, slope −1.7
-16
-14
-12
-10
-8
-6
-4
-2
0
-2 0 2 4 6 8 10
log 1
0 ⟨P
DF
⟩
log10 ρ/ρ0
t=0t=0.26tfft=0.42tff
-1.695(2)-0.999(5)
lognormal
• Initial conditions, t = 0; First subgrids created, t = 0.26tff; Deep AMR hierarchy, t = 0.42tff
• Effective linear resolution: 5×105 (5 pc – 2 AU)
• Two breaks in slope: at ρ ∼ 106.2ρ0 (power index −1.5) and at 107ρ0 (power index −1)
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Density PDFs for subvolumes around cores
49
Three selected condensed objects and their PDFs; ∼ (700 AU)3
-5
-4
-3
-2
-1
0
3 4 5 6 7 8 9
log
10 ⟨
PD
F⟩
log10 ρ/ρ0
core 012-1.25(3)
-6
-5
-4
-3
-2
-1
0
3 4 5 6 7 8 9 10
log
10 ⟨
PD
F⟩
log10 ρ/ρ0
core 008-1.48(5)
-6
-5
-4
-3
-2
-1
0
3 4 5 6 7 8 9 10
log
10 ⟨
PD
F⟩
log10 ρ/ρ0
core 004-1.73(4)
• Individual slopes vary from −1.2 to −1.8
• Rotation-induced pile-ups at ρ > 107ρ0
• Cores 1 and 2 exhibit strong rotation, core 3 shows only modest flattening
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Density PDF and self-similar collapse solutions
50
The PDF for a spherically symmetric configuration with ρ = ρ0(r /r0)−n density profile is a
power law
dV = 4
3πr 3
0 d
[
(
ρ
ρ0
)−3/n]
∝ d(
ρ−m)
. (5)
The projected density of an infinite sphere with the ρ ∼ r−n density profile,
Σ(R) = 2
∫∞
0ρ
(√
R2 + x2)
d x ∝ R1−n , (6)
also has a power-law PDF,
dS ∝ d(
Σ− 2
n−1
)
∝ d(
Σ−p
)
. (7)
For the LP [Larson-Penston, 1969] , PF [Penston, 1969], and EW [Shu, 1977] similarity
solutions: n = 2, 127
, and 32
; m = 32
, 74
, 2; p = 2, 2.8, and 4, respectively.
Kritsuk, Norman & Wagner (2011)
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Projected density PDF from AMR simulation
51
Extended power law tail: > 2 dex in density, slope −2.5
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
-1 0 1 2 3 4
log 1
0 P
DF
log10 Σ/⟨Σ⟩
t=0t=0.43tff-2.50(3)
lognormal
• Initial conditions, t = 0; Deep AMR hierarchy, t = 0.43tff
• Effective linear resolution: 5×105 (5 pc – 2 AU)
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Summary: self-gravity
52
Star-forming molecular clouds develop power-law tails at the high end of
the density PDF.
We attribute the origin of the tails to the fundamental self-similar properties
of the r−2 isothermal collapse and r−12/7 pressure-free collapse laws that
control the density profiles of collapsing structures.
Power-law indices for the mass density PDF: m ∈ [−7/4,−3/2]
Power-law indices for the projected density PDF: m ∈ [−2.8,−2]
Excellent agreement of simulations with (semi-)analytical solutions and
with observations
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Outline
53
IV. Numerics
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
KITP-2007 code comparison project
54
High-order accurate numerical methods for compressible MHD turbulence
simulation
Numerical stability is often an issue in supersonic regime
10 methods for ideal MHD (9 grid-based schemes and 1 SPH)
ENZO, FLASH, K-T, L&L, Pluto, PPML, Ramses, Stagger, Zeus, and
PHANTOM-SPH
Isothermal turbulence decay problem from initial conditions generated with
the Stagger code
Initial rms sonic Mach number, Ms = 9; Alfvén Mach number, MA = 4.5
Resolution: 2563, 5123, and 10243 (for a few codes)
Decay is followed for 4 dynamical times, t ∈ [0,0.2]
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Supersonic MHD turbulence decay
55
Evolution of kinetic energy density, 5123
0.6
0.8
1
1.2
1.4
1.6
0 0.05 0.1 0.15 0.2
log 1
0 <
u2 >/2
t
ENZOFLASH
KT-MHDLL-MHDPLUTO
PPMLRAMSES
STAGGERZEUS
• All grid-based codes agree
• EK is determined mostly by the large-scale flow (steep power spectra of velocity) ⇒ all
codes resolve the large “eddies” sufficiently well
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Supersonic MHD turbulence decay
56
Magnetic energy density, 5123
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0 0.05 0.1 0.15 0.2
log 1
0 <
B2 >
/2
t
ENZOFLASH
KT-MHDLL-MHDPLUTO
PPMLRAMSES
STAGGERZEUS
• Grid codes maintain different levels of magnetic energy witin a factor of ∼ 1.4
• FLASH and PPML preserve the highest levels of EM
• Stagger shows the lowest levels of EM
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Supersonic MHD turbulence decay
57
Compensated velocity power spectra: 5123, t = 0.4tdyn
-1
-0.5
0
0.5
1
1.5
2
0 0.5 1 1.5 2 2.5
log 1
0 k5/
3 P(u
, k)
log10 k/kmin
ENZOFLASH
KT-MHDLL-MHDPLUTO
PPMLRAMSES
STAGGERZEUS
PHANTOM
• Stagger shows an outstanding result!
• PPML is consistently good
• Phantom-SPH is the most diffusive
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Supersonic MHD turbulence decay
58
Magnetic energy spectra: 5123, t = 0.4tdyn
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0 0.5 1 1.5 2 2.5
log 1
0 P
(B, k
)
log10 k/kmin
ENZOFLASH
KT-MHDLL-MHDPLUTO
PPMLRAMSES
STAGGERZEUS
PHANTOM
• PPML and FLASH preserve more power at high wavenumbers
• Phantom-SPH manages to dissipate nearly all magnetic energy
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Supersonic MHD turbulence decay
59
Compensated velocity power spectra: 2563, t = 0.4tdyn
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
P(u
,k)/
Pre
f(u,k
)
log10 k/kmin
ENZOFLASH
KT-MHDLL-MHDPLUTO
PPMLRAMSES
STAGGERZEUS
PHANTOM75%
reference
• Stagger, PPML, and Ramses show the best effective bandwidth
• Phantom-SPH at 5123 particles is the most diffusive method
• Reference solution is 10243 PPML data filtered down to 2563
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Supersonic MHD turbulence decay
60
Compensated spectra of magnetic energy: 2563, t = 0.4tdyn
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
P(B
,k)/
Pre
f(B,k
)
log10 k/kmin
ENZOFLASH
KT-MHDLL-MHDPLUTO
PPMLRAMSES
STAGGERZEUS
PHANTOM75%
reference
• FLASH and PPML are the best
• Effective bandwidth of FLASH is ∼ 3.2 times larger than that of Stagger
• Phantom-SPH quickly dissipates ∼ 80% of magnetic energy ⇒ useless for this application
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Supersonic MHD turbulence decay
61A relative measure of the Reynolds number, 5123
0
0.5
1
1.5
2
0 0.05 0.1 0.15 0.2
2Ω +
4/3
∆
t
ENZOFLASH
KT-MHDLL-MHDPLUTO
PPMLRAMSES
STAGGERZEUS
• Stagger shows an outstanding result
• PPML follows
• ZEUS shows the lowest effective Re
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Supersonic MHD turbulence decay
62A relative measure of the magnetic Reynolds number, 5123
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.05 0.1 0.15 0.2
J2
t
ENZOFLASH
KT-MHDLL-MHDPLUTO
PPMLRAMSES
STAGGERZEUS
• PPML and FLASH show the highest Rem
• Stagger, LL-MHD, and ENZO show the lowest Rem
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Supersonic MHD turbulence decay
63Dilatational-to-solenoidal ratio, 5123, t = 0.4tdyn
-0.8
-0.6
-0.4
-0.2
0
0.2
0 0.5 1 1.5 2 2.5
log 1
0 χ(
k)
log10 k/kmin
ENZOFLASH
KT-MHDLL-MHDPLUTO
PPMLRAMSES
STAGGERZEUS
• KT-MHD and ENZO show substantially higher power in dilatational modes at Nyquist k
• Stagger also shows an excess at intermediate-high wavenumbers
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Supersonic MHD turbulence decay
64Dilatational-to-solenoidal ratio: 5123, t = 4tdyn
-0.8
-0.6
-0.4
-0.2
0
0.2
0 0.5 1 1.5 2 2.5
log 1
0 χ(
k)
log10 k/kmin
ENZOFLASH
KT-MHDLL-MHDPLUTO
PPMLRAMSES
STAGGERZEUS
• Late evolution with KT-MHD, ENZO, and Stagger is substantially affected in a wide k-range
• Extent of the inertial range is reduced for KT-MHD, ENZO, and Stagger
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Summary: Numerics65
Development of consistently stable higher-order accurate schemes for
MHD with shocks is desired
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Perspective66
Self-gravity (deep AMR; Sink particles)
High Resolution MHD (BIG uniform grids; AMR)
Helicity, dynamo (Shearing box; Helical forcing)
Non-ideal effects
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk
Acknowledgments67
This research was supported in part by the National Science Foundation
through grants AST-0607675, AST-0808184, and AST-0908740, as well as
through TeraGrid resources provided by NICS and SDSC (MCA07S014) and
through DOE Office of Science INCITE-2009 and DD-2010 awards allocated at
NCCS (ast015/ast021).
MIST LANL Astro Seminar – December 20, 2010Alexei Kritsuk