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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 (V ázquez-Semadeni 1997) Large number of degrees of freedom

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Turbulence l.jpg

Turbulence

14 April 2003

Astronomy G9001 - Spring 2003

Prof. Mordecai-Mark Mac Low


General thoughts l.jpg
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 l.jpg
Incompressible Turbulence

  • Incompressible Navier-Stokes Equation

  • No density fluctuations:

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

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


Dissipation l.jpg

Lesieur 1997

Dissipation


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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-2P

  • 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


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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


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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


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Protostellar Outflows

  • Fraction of mass accreted fwis 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 ηwfrom turbulence. Assume momentum conservation (McKee 89),


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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


Magnetorotational instabilities l.jpg
Magnetorotational Instabilities

  • Application of Balbus-Hawley (1992,1998) instabilities to galactic disk by Sellwood & Balbus (1999)

MMML, Norman, Königl, Wardle 1995


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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


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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)


Stellar winds l.jpg
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


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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


H ii region energy input l.jpg
HIIRegion 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 l.jpg
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


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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 l.jpg
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 aLandaR, 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

gravity or

other body

forces

conserved

variables

fluxes

pressure


Godunov method l.jpg
Godunov method

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


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Characteristic averaging

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

tn+1

tn+1

or

tn

tn

xj

xj

xj-1

xj+1

xj-1

xj+1

subsonic

flow

supersonic

flow

(from left)


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Characteristic speeds

  • Characteristic speeds are not constant across rarefaction or shock because of change in pressure


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Riemann problem

  • A typical analytic solution for pressure (P. Ricker) is given by the root of


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