MHD Simulations of Interstellar Turbulenceakpc.ucsd.edu/Multiphase/LosAlamos2010.pdf · MHD...

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

http://akpc.ucsd.edu

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]


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].


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].


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]


Young stars and molecular gas in Taurus

8

Goldsmith et al. (2007)


Outline

9

I. Scaling, intermittency,and compressible cascade


ENZO simulation, 2008

10


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]


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]


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)


S⊥2 = 4

3S∥

2 converts into S⊥2 ≈ 1.27S∥

2 at Ms = 6 [Kritsuk et al. 2007]


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)


S∥3(uuu,ℓ) does not scale linearly with ℓ at Ms = 6 [Kritsuk et al. 2007]


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.]


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); . . . ]


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)


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

)


Outline

19

II. Effects of magnetic field and EOS


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


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)]


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)


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 →


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


http://akpc.ucsd.edu/Multiphase/mpB02_512px.mpeg

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


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


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


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


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)


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


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)


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


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


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


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


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]


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


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


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


Outline

41

III. Effects of self-gravity


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.


Simulated filamentary stucture of molecular clouds

43

.

Rendering : NCSA Advanced Scientific Visualization Laboratory: Donna Cox, Matt Hall, AJ Christensen,

Bob Patterson, Stuart Levy


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)


http://akpc.ucsd.edu/Isothermal/wide2medium_11-17.mov

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


Column density PDFs: Star-forming MCs

46

Kainulainen et al. (2009)


Column density PDFs: Non-star-forming MCs

47

Kainulainen et al. (2009)


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)


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


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)


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)


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


Outline

53

IV. Numerics


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]


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



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



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



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



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



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



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



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



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



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


Summary: Numerics65

Development of consistently stable higher-order accurate schemes for

MHD with shocks is desired


Perspective66

Self-gravity (deep AMR; Sink particles)

High Resolution MHD (BIG uniform grids; AMR)

Helicity, dynamo (Shearing box; Helical forcing)

Non-ideal effects


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).


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