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  

t

v



v



v

 

P



v

advective term (nonlinear) viscosity • No density fluctuations:  

v

 0 • No magnetic fields, cooling, gravity, other ISM physics

Dimensional Analysis

• Strength of turbulence given by ratio of advective to dissipative terms, known as

Reynold’s number

Re 

v

    2

v v



VL

 • Energy dissipation rate 

    

L

3 2 

MV

3

L

Lesieur 1997

Dissipation

Incompress ible Navier Stokes equation in component notation 

u i



t



u j



u i



x j

  

p



x i

   2

u i

multiply 1 2  

t u i

2  by

u i u j

1 2  

x j

incompress ibility

u i

2   

u i



u i



p



x i



x i

 

u i

 2

u i

 0 , so the second term can be rewritten as 1 2  

x j u j u i

2

1 2  

t u i

2  1 2  

x j u j u i

2  

u i



p



x i

 

u i

 2

u i

summing 1 2  

t



u

2  on the index

i

1 2  

x j u j



u

2  

u i



p



x i

  

u

  2 

u

averaging over a homogeneou s flow, only scalars survive d dt   1 2 

u

2     

u

  2 

u

   2 where the variance of the vorticity  2 2 is the enstrophy

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) E

is the energy spectrum defined by  1 2 

v

k

2 3

d k

 1 2  0  • Energy spectrum is Fourier transform of auto-correlation function

C

  1 2 

r



Kolmogorov-Obukhov Cascade

• Energy enters at large scales and dissipates at small scales, where 

2 v

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.

The energy spectrum

E



dE dk



C

 

k

 Search for   and  such that the expression has the correct dimension

k



 

 1 ;

E

   

 3   

L

   

T

 2  2 ;  

L T

:  3 The solution is then   2 3 , 

E



C

 2 3

k

 5 3   5 3 , so

Compressibility

• Again examining the Navier-Stokes equation, we can estimate isothermal density fluctuations 

ρ = c s -2



P

• Balance pressure and advective terms: 

P

 

v

 

v



c s

2    

V

2

L

  

U

2 

M

2

c s

2 • Flow no longer purely solenoidal ( 

v

 0).

– Compressible and rotational energy spectra distinct – Compressible spectrum

E c (k) ~ k -2

: Fourier transform of shocks

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 R height H = 200 pc g =15 kpc, and scale • 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

v esc

• Fraction of mass accreted

f w

wind. Shu et al. (1988) is lost in jet or suggest

f w

~ 0.4

 • Mass is ejected close to star, where (2 / ) 1/ 2   195 km s  1  

M

/ M  

R

/10 R • Radiative cooling at wind termination shock steals energy

η w

from turbulence. Assume momentum conservation (McKee 89),   1/ 2 

w



v rms v w



0.01

 

v rms

2 km s

 1    

200 km s

 1

v w

  

Outflow energy input

• Take the surface density of star formation  * in the solar neighborhood (McKee 1989)     9 -2 -1 • Then energy from outflows and jets is

e

   1 2

f w



w

 *

v w

2

H

 29 -3 erg cm s -1    200 pc

H

   

f w

0.4

 

v w

200 km s -1  

v rms

2 km s -1     *   

Magnetorotational Instabilities

MMML, Norman, Königl, Wardle 1995 • Application of Balbus-Hawley (1992,1998) instabilities to galactic disk by Sellwood & Balbus (1999)

MRI energy input

e

• Numerical models by Hawley, Gammie & Balbus (1995) suggest Maxwell stress tensor 

T R

 

0.6

B

2

e T R

 Way,  29 -3

erg cm s

-1   

B



(220 Myr)

 1  

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-

e



G

(   29

H

)  2  -3 erg cm s -1



200 pc

H

gravitating gas disk (neglecting removal of   

gas

10 M pc -2   100 pc    2 (220 Myr) -1 gas by star formation).

 • They estimate Newton stress energy input (requires unproven positive correlation between radial, azimuthal gravitational forces)

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

e

  24 -1

erg s cm

-3 • 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) 

p

to find average momentum input



260 km s -1



M

 where mean mass/cluster   

M



N H

 22 -2     3 /14   

M cl

6 10 M    1/14  440 M , and

N H

varies weakly

H

II

Region Energy Input

• The number of OB associations driving H II

e

regions in the Milky Way is about

N OB =

650 (from McKee & Williams 1997 with S 49 >1)  • Need to assume

v ion =

10 km s -1 , and that star formation lasts for about

t ion

=18.5 Myr, so: 

p N v OB ion e

2  2

R Ht g ion

   

M

 440 M  30 -1 erg s cm -3    

N H

 22 -2     3 / 14   

M cl

6 10 M    1 / 14      

N OB

650

v ion

 10 km s -1   

H

200 pc     1   

R g

15 kpc     2   

t ion

18.5 Myr     1

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 σ SN =1 SNu ( Capellaro et al. 1999 ), so the SN rate is 1/50 yr • Fraction of energy surviving radiative cooling η SN ~ 0.1 (Thornton et al. 1998)

Supernova Energy Input

• If we distribute the SN energy equally over a galactic disk,

e

  

SN



SN E

2

R H g SN





 26 -1 erg s cm -3





SN



SN

0.1

1 SNu   200 pc

H

   

R g

15 kpc     2 

E SN

51 10 erg   • SNe appear hundreds or thousands of times more powerful than all other energy sources

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 

a



t



u



a

   0 ;

a

(  • Zone average values must satisfy

a j n

 1     

j j

 1  1 2 2

a

, 0 )  

d

 

a

0  

j

• A piecewise continuous function with a parabolic profile in each zone that does so is

a



a L

,

j



x

 

a j



a

6 ,

j

 1 

x

  ;

x

     

j

 1

j

2 

a j



a R

,

j



a L

,

j

;

a

6  6

a n j

 1 2 

a L

,

j



a R

,

j

   

Interpolation to zone edges

• To find the left and right values

a L

and

a R

compute a polynomial using nearby zone ,

a L a R

if

a L

,

j

 3

a n j

 2

a R

,

j

if



a R,j



a L

,

j



a L a n j

 1 2



a L

,

j

_

a R

,

j

 

a R

,

j



a L

,

j



2 6 • And similarly for

a R,j

to force montonicity.

Conservative Form

• Euler’s equations in conservation form on a 1D

U

Cartesian grid         

u



v



E

      ,

F

 

U

      

t

   

u



F u



x



uv uE

 2 

up

  

H



x

    ,

H

 

G

      0

p

0     ,

G

0         0

g

0   

ug

      conserved variables fluxes pressure gravity or other body forces

Godunov method

• Solve a Riemann shock tube problem at every zone boundary to determine fluxes

x j-1

Characteristic averaging

• To find left and right states for Riemann problem, average over regions covered by characteristic: max(

c s ,u

)

Δt

t n+1 t n+1 or t n x j t n x j x j+1 x j-1 x j+1 subsonic flow supersonic flow (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

f L



P

,

U L





R



P

,

U R





u R

where, with

I

  (

L

,

R

)

u L

 0 ,

f I



P

,

U I



  



P

     2

c s

 ,

I

1      

P I



P I

 

P P



A I



B I

 1   2     1    

A I





  2 1





I

if

P



P I

(shock) if

P



P I

(rarefacti on)

B I

    1  1

P I