Turbulence

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Turbulence

• 14 April 2003
• Astronomy G9001 - Spring 2003
• Prof. Mordecai-Mark Mac Low

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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

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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
  Reynolds number
• Energy dissipation rate

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Dissipation
Lesieur 1997
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Fourier Power Spectrum

• Homogeneous turbulence can be considered in
  Fourier space, to look at structure at different
  length scales L 2p/k
• Incompressible turbulent energy is just v2
• 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
• Reynolds 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 e must be constant at
  all scales, and spectrum depends on k and e.

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Compressibility

• Again examining the Navier-Stokes equation, we
  can estimate isothermal density fluctuations ??
  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 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 ?w from 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

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Magnetorotational Instabilities
MMML, Norman, Königl, Wardle 1995

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

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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)

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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

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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

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H II Region Energy Input

• The number of OB associations driving H II
  regions in the Milky Way is about NOB650 (from
  McKee Williams 1997 with S49gt1)
• Need to assume vion10 km s-1, and that star
  formation lasts for about tion18.5 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
  sSN1 SNu (Capellaro et al. 1999), so the SN rate
  is 1/50 yr
• Fraction of energy surviving radiative cooling
  ?SN 0.1 (Thornton et al. 1998)

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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 Glovers 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.

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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 ??j
• In some cases this is not monotonic, so add
• And similarly for aR,j to force montonicity.

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Conservative Form

• Eulers equations in conservation form on a 1D
  Cartesian grid

gravity or other body forces
conserved variables
fluxes
pressure
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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) ?t

tn1
tn1
or
tn
tn
xj
xj
xj1
xj-1
xj1
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