What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Turbulence in Astrophysics
Wolfram Schmidt
Workshop on Turbulence and Hydrodynamical Instabilities
Garching, 17-19 November 2008
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Overview
1. What do we mean by turbulence in astrophysics?
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Overview
1. What do we mean by turbulence in astrophysics?
2. Computing astrophysical turbulence
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Overview
1. What do we mean by turbulence in astrophysics?
2. Computing astrophysical turbulence
3. What can we learn from numerical simulations?
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Velocity Dispersion
Inferred from Doppler line broadening
I Absorption lines of stars
I Molecular (CO) emission lines
I HI emission lines
Convection in stellar atmospheres .
I X-ray emission, Faraday rotation maps
Turbulent inter-cluster medium .
Wolfram Schmidt
Supersonic molecular cloud turbulence .
Turbulence in the interstellar medium .
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Velocity Dispersion
Inferred from Doppler line broadening
I Absorption lines of stars
I Molecular (CO) emission lines
I HI emission lines
Convection in stellar atmospheres .
I X-ray emission, Faraday rotation maps
Turbulent inter-cluster medium .
Supersonic molecular cloud turbulence .
Turbulence in the interstellar medium .
Reynolds number
Lhδv i
Re =
ν
I Athlete swimming ∼ 106
I Blue Whale ∼ 108
I Queen Elizabeth 2 ∼ 109
I Sun ∼ 1014
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Power Laws Sp (l) ∝ l ζp for Molecular Clouds
I δv (l) from PCA decompositions of
12 CO imaging (Brunt & Heyer 2004)
I Brunt & Heyer (2002) found
γ = ζ2 /2 ≈ 0.33 . . . 0.81
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Power Laws Sp (l) ∝ l ζp for Molecular Clouds
I δv (l) from PCA decompositions of
12 CO imaging (Brunt & Heyer 2004)
I Brunt & Heyer (2002) found
γ = ζ2 /2 ≈ 0.33 . . . 0.81
Wolfram Schmidt
I Structure functions Sp (l) := hδv p (l)i
from line centroid velocities in
Polaris (Hily-Blant et al. 2008)
I Z2 := ζ2 /ζ3 ≈ 0.7
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Universality of Turbulence?
FLASH3 10243 simulation of supersonic isothermal turbulence with
solenoidal forcing (Federrath, Klessen & Schmidt 2008)
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Universality of Turbulence?
FLASH3 10243 simulation of supersonic isothermal turbulence with
compressive forcing (Federrath, Klessen & Schmidt 2008)
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Scaling Properties of Supersonic Turbulence
Velocity structure functions reveal different power laws depending
on the large-scale forcing (Schmidt, Federrath & Klessen 2008)
S¦p
104
1000
100
10
1
0.5
1.0
5.0
Wolfram Schmidt
10.0
50.0 100.0
S3¦
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Scaling Properties of Supersonic Turbulence
Z ¦p
1.4
1.2
comp
1.0
0.8
sol
0.6
1
2
3
4
5
p
I Scaling laws deviate largely from K41
and the She-Lévêque model (Kritsuk
et al. 2007, Schmidt et al. 2008)
I Compressive forcing produces
scalings different from Boldyrev 2002
I Zp can be fitted by generalized
log-Poisson intermittency models
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Scaling Properties of Supersonic Turbulence
Z ¦p
1.4
Z p¦
1.4
1.2
1.2
comp
1.0
1.0
comp
0.8
0.8
sol
sol
0.6
0.6
1
2
3
4
5
p
1
I Scaling laws deviate largely from K41
and the She-Lévêque model (Kritsuk
et al. 2007, Schmidt et al. 2008)
I Compressive forcing produces
scalings different from Boldyrev 2002
I Zp can be fitted by generalized
log-Poisson intermittency models
Wolfram Schmidt
2
3
4
5
p
I Kritsuk et al. 2007: two-point
statistics of mass-weighted velocity
ρ1/3 v for compressible turbulence
I Nearly universal scaling exponents
for S̃p (l) := hδ(ρ1/3 v )p (l)i (Schmidt,
Federrath & Klessen 2008)
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Large Eddy Simulation (LES)
In astrophysics: numerical resolution physical dissipation scale
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Large Eddy Simulation (LES)
In astrophysics: numerical resolution physical dissipation scale
Finite-volume schemes:
I
grid discretization errors
I
energy flux from resolved to subgrid scales SGS
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Large Eddy Simulation (LES)
In astrophysics: numerical resolution physical dissipation scale
Finite-volume schemes:
I
grid discretization errors
I
energy flux from resolved to subgrid scales SGS
Most common in astrophysics: ILES or SPH
I
numerical solution is interpreted as smoothed approximation
I
energy flux is implicitly modelled by numerical dissipation
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Subgrid scale model
Additional terms in the compressible Euler equations:
I
I
I
1/2
Turbulence stress: τij∗ = 2Cν ∆ksgs Sij∗ (eddy-viscosity closure)
Turbulence energy flux: Σsgs = τij Sij
Turbulence pressure: Psgs = 23 ρksgs → Peff = 1 + 13 M2sgs P
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Subgrid scale model
Additional terms in the compressible Euler equations:
I
I
I
1/2
Turbulence stress: τij∗ = 2Cν ∆ksgs Sij∗ (eddy-viscosity closure)
Turbulence energy flux: Σsgs = τij Sij
Turbulence pressure: Psgs = 23 ρksgs → Peff = 1 + 13 M2sgs P
Balance law for the unresolved turbulence energy ksgs
(Schumann 1975, Schmidt et al. 2006):
“
”
D
1
1/2
ksgs − ∇ · ρCκ ∆ksgs ∇ksgs =
Dt
ρ
„
«
3/2
ksgs
2
1/2
Cν ∆ksgs |S ∗ |2 −
+ Cλ ksgs d − C
3
∆
Resolved kinetic energy
Compression effects
I
Turbulence cascade
SGS turbulence energy
Dissipation
Internal energy
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Subgrid scale model
Fluid dynamical effects
I
physical model for energy flux
I
controlling the bottleneck effect
I
instabilities
I
turbulence pressure and energy budget
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Subgrid scale model
Fluid dynamical effects
I
physical model for energy flux
I
controlling the bottleneck effect
I
instabilities
I
turbulence pressure and energy budget
Parametrizations
I
turbulent transport (e.g. flame propagation)
I
turbulence-regulated processes (e.g. star formation)
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Example: LES of Thermonuclear Supernovae
1/2
Turbulent flame speed ∝ ksgs (Niemeyer & Hillebrandt 1995)
Snapshot form a high-resolution simulation
0.6 sec after ignition (Röpke et al. 2007)
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Example: LES of Thermonuclear Supernovae
1/2
Turbulent flame speed ∝ ksgs (Niemeyer & Hillebrandt 1995)
Snapshot form a high-resolution simulation
0.6 sec after ignition (Röpke et al. 2007)
Wolfram Schmidt
Structure functions in the burning region
(Ciaraldi-Schoolmann et al. 2008)
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Adaptive mesh refinement (AMR)
I
Usually applied to follow gravitational collapse (e.g. Abel,
Bryan and Norman 2002)
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Adaptive mesh refinement (AMR)
I
Usually applied to follow gravitational collapse (e.g. Abel,
Bryan and Norman 2002)
I
Potential applicability to turbulence because of intermittency
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Adaptive mesh refinement (AMR)
I
Usually applied to follow gravitational collapse (e.g. Abel,
Bryan and Norman 2002)
I
Potential applicability to turbulence because of intermittency
I
Pioneering AMR simulations of supersonic turbulence by
Kritsuk et al. 2006
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Adaptive mesh refinement (AMR)
I
Usually applied to follow gravitational collapse (e.g. Abel,
Bryan and Norman 2002)
I
Potential applicability to turbulence because of intermittency
I
Pioneering AMR simulations of supersonic turbulence by
Kritsuk et al. 2006
I
Refinement criteria based on the turbulence control variables
enstrophy and rate of compression (Schmidt et al. 2008)
1
Dd
1
h1 = ω 2 , and h2 = −
'
|S|2 − ω 2 + cs2 ∇2 ln ρ
2
Dt
2
Thresholds given by statistical moments rather than local
normalizations
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Example: Compressively driven supersonic turbulence
Enzo simulation with 1923 root grid and 1 level of refinement
(resolution factor 4)
mass density slice
Mach number slice
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Example: Compressively driven supersonic turbulence
Comparison to 7683 static grid simulation
mass density PDFs
vorticity modulus PDFs
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Turbulent Wake of a Moving subcluster
Enzo simulation with 643 root grid and 2 levels of refinement by
enstrophy and compression (based on Iapichino et al. 2008)
AMR
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Turbulent Wake of a Moving subcluster
Enzo simulation with 643 root grid and 2 levels of refinement by
enstrophy and compression (based on Iapichino et al. 2008)
AMR + SGS model
AMR
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Adaptively Refined Large Eddy Simulation
vorticity slice
SGS turbulence energy flux
SGS turbulence energy
Shear-Improved Technique
Treatment of inhomogeneous turbulence (Lévêque et al. 2007,
Schmidt & Lévêque in prep.):
1/2
Σsgs = 2Cν ∆ksgs
(Sij − hSij i)∗ Sij
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Cosmological Simulations of Galaxy Clusters
Enzo simulation with nested 1283 root grid and 4 levels of
refinement by overdensity, flat CDM background, 1283 particles
(Maier et al. in prep.)
mass density slice
SGS turbulence energy slice
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Cosmological Simulations of Galaxy Clusters
Enzo simulation with nested 1283 root grid and 4 levels of
refinement by overdensity, flat CDM background, 1283 particles
(Maier et al. in prep.)
mass density slice
SGS turbulence energy slice
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
The Role of Small-Scale Turbulence in Galaxy Clusters
I
The spatial distribution of SGS turbulence energy traces the
merging history
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
The Role of Small-Scale Turbulence in Galaxy Clusters
I
The spatial distribution of SGS turbulence energy traces the
merging history
I
The peak of turbulence energy is found around 0.5 virial radii
200
Sarkar SGS
turbulent velocity (scaled) [km/s]
180
160
140
120
100
80
60
0.1
1.0
r (Rvir)
1/2
Radial profile of ksgs
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
The Role of Small-Scale Turbulence in Galaxy Clusters
I
The spatial distribution of SGS turbulence energy traces the
merging history
I
The peak of turbulence energy is found around 0.5 virial radii
I
Polytropic index closer to isothermal (Pratt & Arnaud 2002)
with plateau at 0.2 virial radii (Vikhlinin et al. 2006)
200
2.0
160
Sarkar SGS
effective polytropic index
turbulent velocity (scaled) [km/s]
No SGS
Sarkar SGS
180
140
120
100
1.5
1.0
0.5
80
60
0.0
0.1
1.0
r (Rvir)
0.1
1.0
r (Rvir)
1/2
Radial profile of ksgs
effective polytropic index
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Star Formation in Disk Galaxies
I
What controls star formtion (gravity, turbulence, magnetic
fields, ...)?
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Star Formation in Disk Galaxies
I
What controls star formtion (gravity, turbulence, magnetic
fields, ...)?
I
Krumholz & McKee 2005: star formation rate (SFR) is a
function of the turbulent Mach number and virial parameter:
SFRff ∝ M−0.32 α−0.68
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Star Formation in Disk Galaxies
I
What controls star formtion (gravity, turbulence, magnetic
fields, ...)?
I
Krumholz & McKee 2005: star formation rate (SFR) is a
function of the turbulent Mach number and virial parameter:
SFRff ∝ M−0.32 α−0.68
I
Parametrization of SFR in disk galaxy simulations in terms of
Msgs = (2ksgs )1/2 /cs
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Disk Galaxy Evolution
Enzo simulation with 8 levels of refinement by overdensity, star
particles, feedback model and turbulence-regulated SFR
(Tasker & Bryan 2006, Hupp et al. in prep.)
20 Myrs
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Disk Galaxy Evolution
Enzo simulation with 8 levels of refinement by overdensity, star
particles, feedback model and turbulence-regulated SFR
(Tasker & Bryan 2006, Hupp et al. in prep.)
20 Myrs
100 Myrs
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Disk Galaxy Evolution
Enzo simulation with 8 levels of refinement by overdensity, star
particles, feedback model and turbulence-regulated SFR
(Tasker & Bryan 2006, Hupp et al. in prep.)
20 Myrs
100 Myrs
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200 Myrs
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
The Kennicutt-Schmidt law
I
Presently, it is not possible to create a grand-design spiral
galaxy just from first principles
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
The Kennicutt-Schmidt law
I
Presently, it is not possible to create a grand-design spiral
galaxy just from first principles
I
Investigate properties such as the Kennicutt-Schmidt law
(Tasker, Bryan & Tan 2008), HI velocity dispersion (Agertz et
al. 2008), or chemical evolution (Dobbs et al. 2008)
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
The Kennicutt-Schmidt law
I
Presently, it is not possible to create a grand-design spiral
galaxy just from first principles
I
Investigate properties such as the Kennicutt-Schmidt law
(Tasker, Bryan & Tan 2008), HI velocity dispersion (Agertz et
al. 2008), or chemical evolution (Dobbs et al. 2008)
Star Formation Data
Kennicutt, R.C. 1998, ApJ 498,541
linear fit
Star Formation Data
Kennicutt, R.C. 1998, ApJ 498,541
linear fit
1.4
1.4
100
ΣSFR [Msol yr-1 kpc-2]
0.1
1
N
10
100
0.8
0.01
0.6
0.001
0.4
1000
CreationTime [Gyr]
1
1
0.1
100
1.2
ΣSFR = A Σ gas
A = 3.671e-03
N = 1.41
10
0.2
1.2
10
1
1
ΣSFR [Msol yr-1 kpc-2]
0.1
0.1
1
10
Σgas [Msol pc-2]
Σgas [Msol pc-2]
turbulence-regulated SFR
SFR with constant efficiency
Wolfram Schmidt
Turbulence in Astrophysics
100
0.8
0.01
0.6
0.001
0.4
1000
0.2
CreationTime [Gyr]
ΣSFR = A ΣNgas
A = 3.193e-04
N = 2.06
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Outlook: Closures for Highly Compressible Turbulence
Correlation between turbulence energy flux and closures (Gauss
filtering of 10243 data on inertial-range length scale 32∆, Schmidt,
Federrath & Kritsuk in prep.)
2000
1500
1000
500
0
-500
-500
0
500
1000
1500
2000
Smagorinksy closure
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Outlook: Closures for Highly Compressible Turbulence
Correlation between turbulence energy flux and closures (Gauss
filtering of 10243 data on inertial-range length scale 32∆, Schmidt,
Federrath & Kritsuk in prep.)
2000
2000
1500
1500
1000
1000
500
500
0
0
-500
-500
-500
0
500
1000
1500
2000
-500
0
500
∗
1000
1500
2000
det S closure (Woodward et al. 2001)
Smagorinksy closure
Wolfram Schmidt
Turbulence in Astrophysics
What do we mean by turbulence in astrophysics?
Computing astrophysical turbulence
What can we learn from numerical simulations?
Outlook: Closures for Highly Compressible Turbulence
Correlation between turbulence energy flux and closures
(Gauss filtering of 10243 data on inertial-range length scale 32∆,
Schmidt, Federrath & Kritsuk in prep.)
2000
2000
1500
1500
1000
1000
500
500
0
0
-500
-500
-500
0
500
1000
1500
2000
-500
0
500
1000
non-linear closure (τ sgs ∼
Smagorinksy closure
Wolfram Schmidt
Turbulence in Astrophysics
1500
2000
O2 (∇
⊗ v))