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

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advective term (nonlinear)

viscosity

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Dimensional Analysis• Strength of turbulence given by ratio of

advective to dissipative terms, known as Reynold’s number

• Energy dissipation rate

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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

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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.

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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

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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

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Outflow energy input

• Take the surface density of star formation in the solar neighborhood (McKee 1989)

• Then energy from outflows and jets is

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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,

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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)

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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

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where mean mass/cluster 440 M , and varies weakly

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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:

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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

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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

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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.

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Conservative Form

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

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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

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or

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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

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