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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

Incompressible Turbulence

- Incompressible Navier-Stokes Equation
- No density fluctuations:
- No magnetic fields, cooling, gravity, other ISM physics

advective term

(nonlinear)

viscosity

Dimensional Analysis

- Strength of turbulence given by ratio of advective to dissipative terms, known as Reynold’s number
- Energy dissipation rate

Dissipation

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

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.

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

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 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),

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

- 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,

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)

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

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:

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

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

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.

Conservative Form

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

gravity or

other body

forces

conserved

variables

fluxes

pressure

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+1

tn+1

or

tn

tn

xj

xj

xj-1

xj+1

xj-1

xj+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

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