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

2. 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

3. Incompressible Turbulence Incompressible Navier-Stokes Equation No density fluctuations: No magnetic fields, cooling, gravity, other ISM physics advective term (nonlinear) viscosity

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

5. Dissipation Lesieur 1997

7. 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

8. 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.

10. Compressibility Again examining the Navier-Stokes equation, we can estimate isothermal density fluctuations  ρ = c s -2  P Balance pressure and advective terms: Flow no longer purely solenoidal (  v  0). –Compressible and rotational energy spectra distinct –Compressible spectrum E c (k) ~ k -2 : Fourier transform of shocks

11. 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

12. 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 g =15 kpc, and scale height H = 200 pc Try to account for initial radiative losses when necessary

13. 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

14. Protostellar Outflows Fraction of mass accreted f w is lost in jet or wind. Shu et al. (1988) suggest f w ~ 0.4 Mass is ejected close to star, where Radiative cooling at wind termination shock steals energy η w from turbulence. Assume momentum conservation (McKee 89),

15. Outflow energy input Take the surface density of star formation in the solar neighborhood (McKee 1989) Then energy from outflows and jets is

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

17. MRI energy input Numerical models by Hawley, Gammie & Balbus (1995) suggest Maxwell stress tensor Energy input, so in the Milky Way,

18. 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

19. 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)

20. 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

21. 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

22. H II Region Energy Input The number of OB associations driving H II 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:

23. 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)

24. 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

25. 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

26. 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.

27. 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

28. Interpolation to zone edges To find the left and right values a L and a R, compute a polynomial using nearby zone averages. For constant zone widths Δξ j In some cases this is not monotonic, so add: And similarly for a R,j to force montonicity.

29. Conservative Form Euler’s equations in conservation form on a 1D Cartesian grid conserved variables fluxespressure gravity or other body forces

30. Godunov method Solve a Riemann shock tube problem at every zone boundary to determine fluxes

31. Characteristic averaging To find left and right states for Riemann problem, average over regions covered by characteristic: max(c s,u) Δt tntn t n+1 x j+1 x j-1 xjxj or tntn t n+1 x j+1 x j-1 xjxj subsonic flow supersonic flow (from left)

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

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