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Turbulence
14 April 2003
Astronomy G9001 - Spring 2003
Prof. Mordecai-Mark Mac Low
General Thoughts
• Turbulence often identified with incompressible turbulence only
• More general definition needed (Vázquez-Semadeni 1997)– Large number of degrees of freedom– Different modes can exchange energy– Sensitive to initial conditions– Mixing occurs
Incompressible Turbulence
• Incompressible Navier-Stokes Equation
• No density fluctuations:
• No magnetic fields, cooling, gravity, other ISM physics
vvvv 2
P
t
advective term (nonlinear)
viscosity
0 v
Dimensional Analysis• Strength of turbulence given by ratio of
advective to dissipative terms, known as Reynold’s number
• Energy dissipation rate
L
MV
T
LME
3
3
2
VL
v
vv2
Re
DissipationLesieur 1997
2
222
2
2
1 asrewritten becan termsecond theso
,0 ibility incompress
2
1
2
1
by multiply
notationcomponent in
equation Stokes-Navier ibleIncompress
ijj
ii
iii
iij
ji
i
iij
ij
i
uux
xu
uux
puu
xuu
t
u
ux
p
x
uu
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u
enstrophy theis 2 vorticity theof variance thewhere
2
1
dt
d
survive scalarsonly flow, shomogeneou aover averaging
2
1
2
1
index on the summing
2
1
2
1
2
222
222
222
uuu
uux
puuu
xu
t
i
uux
puuu
xu
t
iij
j
iii
iijj
i
Fourier Power Spectrum• Homogeneous turbulence can be considered
in Fourier space, to look at structure at different length scales L = 2π/k
• Incompressible turbulent energy is just |v|2
• E(k) is the energy spectrum defined by
• Energy spectrum is Fourier transform of auto-correlation function
2 3
0
1 1
2 2kE d k E k dk
v
2
1C
r u x u x r
Kolmogorov-Obukhov Cascade
• Energy enters at large scales and dissipates at small scales, where 2v most important
• Reynold’s number high enough for separation of scales between driving and dissipation
• Assume energy transfer only occurs between neighboring scales (Big whirls have little whirls, which feed on their velocity, and little whirls have lesser whirls, and so on to viscosity - Richardson)
• Energy input balances energy dissipation• Then energy transfer rate ε must be constant at
all scales, and spectrum depends on k and ε.
3532
2332
32231
so ,3
5 ,
3
2 then issolution The
; ;
:dimensioncorrect thehas expression
thesuch that and for Search
spectrumenergy The
kCkE
TLTL
TLTLkELk
kCdkdEkE
Compressibility
• Again examining the Navier-Stokes equation, we can estimate isothermal density fluctuations ρ = cs
-2P• Balance pressure and advective terms:
• Flow no longer purely solenoidal (v 0).– Compressible and rotational energy spectra distinct
– Compressible spectrum Ec(k) ~ k-2: Fourier transform of shocks
22
2
22
Mc
U
LVcP
s
s
vv
Some special cases
• 2D turbulence– Energy and enstrophy cascades reverse– Energy cascades up from driving scale, so large-
scale eddies form and survive– Planetary atmospheres typical example
• Burgers turbulence– Pressure-free turbulence– Hypersonic limit– Relatively tractable analytically– Energy spectrum E(k) ~ k-2
What is driving the turbulence?
• Compare energetics from the different suggested mechanisms (Mac Low & Klessen 2003, Rev. Mod. Phys., on astro-ph)
• Normalize to solar circle values in a uniform disk with Rg =15 kpc, and scale height H = 200 pc
• Try to account for initial radiative losses when necessary
Mechanisms
• Gravitational collapse coupled to shear• Protostellar winds and jets• Magnetorotational instabilities• Massive stars
– Expansion of H II regions– Fluctuations in UV field– Stellar winds– Supernovae
Protostellar Outflows
• Fraction of mass accreted fw is lost in jet or wind. Shu et al. (1988) suggest fw ~ 0.4
• Mass is ejected close to star, where
• Radiative cooling at wind termination shock steals energy ηw from turbulence. Assume
momentum conservation (McKee 89),1
1
200 km s0.01
2 km srms rms
ww w
v v
v v
1/ 2 1/ 21/ 2 1(2 / ) 195 km s / M /10 Rescv GM R M R
Outflow energy input
• Take the surface density of star formation in the solar neighborhood (McKee 1989)
• Then energy from outflows and jets is
*
2*
-3 -129
*-1 -1
1
2200 pc
4 10 erg cm s
0.4 200 km s 2 km s
w w w
w w rms
e f vH
H
f v v
9 -2 -14.5 10 M pc yr
Magnetorotational Instabilities
• Application of Balbus-Hawley (1992,1998) instabilities to galactic disk by Sellwood & Balbus (1999)
MMML, Norman, Königl, Wardle 1995
MRI energy input
• Numerical models by Hawley, Gammie & Balbus (1995) suggest Maxwell stress tensor
• Energy input , so in the Milky Way,
20.6 /(8 )RT B
Re T
2
-3 -11
293 10 erg cm s3 G (220 Myr)
Be
Gravitational Driving
• Local gravitational collapse cannot generate enough turbulence to delay further collapse beyond a free-fall time (Klessen et al. 98, Mac Low 99)
• Spiral density waves drive shocks/hydraulic jumps that do add energy to turbulence (Lin & Shu, Roberts 69, Martos & Cox).
• However, turbulence also strong in irregular galaxies without strong spiral arms
Energy Input from Gravitation• Wada, Meurer, & Norman
(2002) estimate energy input from shearing, self-gravitating gas disk (neglecting removal of gas by star formation).
• They estimate Newton stress energy input (requires unproven positive correlation between radial, azimuthal gravitational forces)
2
2 2
-2 -1
2
29 -3 -1 200 pc
10 M pc 100 pc (220 Myr)
( / )
1 10 erg cm s
gas
gas
H
e G H
Stellar Winds
• The total energy from a line-driven stellar wind over the lifetime of an early O star can equal the energy of its final supernova explosion.
• However, most SNe come from the far more numerous B stars which have much weaker stellar winds.
• Although stellar winds may be locally important, they will always be a small fraction of the total energy input from SNe
H II Region Expansion• Total ionizing radiation (Abbott 82) has energy
• Most of this energy goes to ionization rather than driving turbulence, however.
• Matzner (2002) integrates over H II region luminosity function from McKee & Williams (1997) to find average momentum input
-1 -3241.5 10 erg s cme
1/143 /14
-122 -2 6
260 km s1.5 10 cm 10 M
where mean mass/cluster 440 M , and varies weakly
H cl
H
N Mp M
M N
H II Region Energy Input• The number of OB associations driving H II
regions in the Milky Way is about NOB=650 (from McKee & Williams 1997 with S49>1)
• Need to assume vion=10 km s-1, and that star formation lasts for about tion=18.5 Myr, so:
1 / 143 / 14
22 -2 6
21 1
-1
30
2
-1 -3
1.5 10 cm 10 M
440 M 650 10 km s 200 pc 15 kpc 18.5 Myr
2
2 10 erg s cm H cl
gOB ion ion
OB ion
g ion
N M
RM N v H t
p N ve
R Ht
e
Supernovae
• SNe mostly from B stars far from GMCs– Slope of IMF means many more B than O stars– B stars take up to 50 Myr to explode
• Take the SN rate in the Milky Way to be roughly σSN=1 SNu (Capellaro et al. 1999), so the SN rate is 1/50 yr
• Fraction of energy surviving radiative cooling ηSN ~ 0.1 (Thornton et al. 1998)
Supernova Energy Input• If we distribute the SN energy equally over
a galactic disk,
• SNe appear hundreds or thousands of times more powerful than all other energy sources
2
-1 -3
2
26
51
2 10 erg s cm0.1 1 SNu
200 pc
15 kpc 10 erg
SN SN SN
g
SN SN
g SN
Ee
R H
R E
H
Assignments
• Abel, Bryan, & Norman, Science, 295, 93 [This will be discussed after Simon Glover’s guest lecture, sometime in the next several weeks]
• Sections 1, 2, and 5 of Klessen & Mac Low 2003, astro-ph/0301093 [to be discussed after my next lecture]
• Exercise 6
Piecewise Parabolic Method
• Third-order advection
• Godunov method for flux estimation
• Contact discontinuity steepeners
• Small amount of linear artificial viscosity• Described by Colella & Woodward 1984, JCP,
compared to other methods by Woodward & Colella 1984, JCP.
Parabolic Advection• Consider the linear advection equation
• Zone average values must satisfy
• A piecewise continuous function with a parabolic profile in each zone that does so is
0)0,( ;0 aaa
ut
a
21
21
1 j
j
daaj
nj
jRjLnjjLjRj
j
jjjjL
aaaaaaa
xxaaxaa
,,6,,
21,6,
2
16 ;
;1
Interpolation to zone edges• To find the left and right values aL and aR,
compute a polynomial using nearby zone averages. For constant zone widths Δξj
• In some cases this is not monotonic, so add:
• And similarly for aR,j to force montonicity.
nj
nj
nj
njjLjRj aaaaaaa 1211,,21 12
1
12
7
6
_2
1 if 23
0 if 2
,,,,,,,
,,,,
jLjRjRjL
njjLR,jjR
njjL
jLnj
njjR
njjRjL
aaaaaaaaaa
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Conservative Form
• Euler’s equations in conservation form on a 1D Cartesian grid
ug
gG
pH
upuE
uv
u
u
F
E
v
uU
Gx
H
x
F
t
U
0
0
,
0
0
0
, ,2
conservedvariables
fluxes pressuregravity orother body forces
Godunov method• Solve a Riemann shock tube problem at every
zone boundary to determine fluxes
Characteristic averaging• To find left and right states for Riemann
problem, average over regions covered by characteristic: max(cs,u) Δt
tn
tn+1
xj+1xj-1xj
or
tn
tn+1
xj+1xj-1xj
subsonicflow
supersonicflow
(from left)
Characteristic speeds• Characteristic speeds are not constant
across rarefaction or shock because of change in pressure
Riemann problem• A typical analytic solution for pressure (P.
Ricker) is given by the root of
III
I
II
Is
II
II
II
LRRRLL
PBA
PPP
Pc
PPBP
APP
UPf
RLI
uuUPfUPf
1
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on)(rarefacti if11
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(shock) if
,
),( with where,
,0,,
2
1
,