Woong-Tae Kim (Harvard-Smithsonian CfA)

1. Magnetorotationally-Driven Turbulence in Astrophysical Disks Woong-Tae Kim (Harvard-Smithsonian CfA)

2. Disks Are Common! • Solid Disks • Planetary rings • Accretion Disks • Protostellar disks • Disks in CVs • Disks in AGN • Galactic Disks

3. Disks Are Common! • Solid Disks • Planetary rings • Accretion Disks • Protostellar disks • Disks in CVs • Disks in AGN • Galactic Disks

4. Disks Are Common! • Solid Disks • Planetary rings • Accretion Disks • Protostellar disks • Disks in CVs • Disks in AGN • Galactic Disks

5. Disks Are Common! • Solid Disks • Planetary rings • Accretion Disks • Protostellar disks • Disks in CVs • Disks in AGN • Galactic Disks Nova Cygni 1992

6. Disks Are Common! • Solid Disks • Planetary rings • Accretion Disks • Protostellar disks • Disks in CVs • Disks in AGN • Galactic Disks

7. Disks Are Common! • Solid Disks • Planetary rings • Accretion Disks • Protostellar disks • Disks in CVs • Disks in AGN • Galactic Disks

8. Accretion Disks • Because of the conservation of the specific angular momentum, the formation of a disk during gravitational infall is unavoidable. • Disks mediate continued accretion, and the release of gravitational binding energy is responsible for luminosity that is observed. • During accretion, a particular element of gas loses angular momentum due to viscosity, slowly flowing towards a black hole at the center. • For radiatively efficient (thin), non-relativistic disks, Frank et al. (1992) • Need to know the behavior of ν(R,t) to close the basic equations. • Under steady conditions, many quantities (e.g., dissipation rate --mass accretion rate, temperature T ~ R-3/4) become independent of ν.

9. α-Disk Models • Steady-state Shakura & Sunyaev (1973)’s α-prescription for viscosit ν= αcsH, with constant α < 1, where H is the disk thickness, assumed to be the maximum scale of the turbulent eddies. Frank et al. (1992) • Subsonic accretion:  vR~ ν/R~ αcsH /R << cs  (cf, cs~10 km s-1; vφ ~103 km s-1) • Although α-disk models do not give complete scalings, they are proven to be very useful; insensitivity of H and Tc to α , UV continuum from CVs.

10. Viscosity • Molecular viscosity: simply too small • ν~ 105 cm2 s-1,l~1010 cm τ ~ l2/ ν~ 3x107 yr, too long for the variability seen in compact object accretion disks. • Remol = lvφ/ν > 1014, molecular viscosity is too weak to bring about dissipation and transport of angular momentum. • Turbulent viscosity may be the cause of the efficient angular momentum transport. • An idea behind α-disk model (ν= vturblturb) • What drives turbulence? • Does the turbulence transport angular momentum outwards?

11. Hydrodynamic Turbulence • Astrophysical disks (Ω ~ R-q with q=0-1.5) with are linearly stable to rotational instability. • No published laboratory experiment for the breakdwon of a Keplerian-like Couette flow (cf., nonlinear Kelvin-Helmholtz instabilities, Triton 1988). • When shear is low or absent as in a boundary layer,  flow laminar shows a nonlinear breakdown to turbulent flow (Hawley, Balbus, & Winters 1999). • Turbulence driven by convectiontransports angular momentum inward with negativeα (Ryu & Goodman 1992; Kley et al. 1993, Stone & Balbus 1996). • Global instabilities (e.g., Papaloizou-Pringle instability) do not generate turbulence (Hawley 1991).

12. Hydromagnetic Turbulence • Accretion disks are magnetized, and by adding new degrees of freedom in their host fluid even very weak magnetic fields can completely alter the stability character of astrophyscial gas. • Rxy couples with Coriolis force • Mxy couples with -dΩ/dlnR • With B-fields, a perturbed fluid element tends to conserve its angular velocity.

13. Magnetorotational Instability • Destabilization of Couette flow by a vertical magnetic field. • Differentially rotating, weakly magnetized disks are locally unstable if • Analyzed first by Velikhov (1959) and Chandrasekhar (1960) for global models; applied to accretion disks by Balbus & Hawley (1991). • Slow MHD waves become unstable. • Growth rate (γmax=0.75Ω ) is so large that the nonlinear stage of the MRI must have important consequences for disks. • Perhaps, no linear instability driven by shear can grow faster.

14. Channel Solutions • As a consequence of MRI, 2D axisymmetric models of weak vertical fields rapidly develop channel solutions. • Goodman & Xu (1994) showed that channel solutions are • the exact nonlinearsolutions of the shearing-sheet MHD equations, and • subject to KH-like parasiticinstabilities in three dimensions (Goodman & Xu 1994). • The parasitic instabilities lead to smaller-scale turbulence in nonlinear stage (Hawley et al. 1995). Hawley & Balbus (1991)

15. 3D MRI • MRI leads to rapid field growth, turbulence, and enhanced transport. • With <B>=0, the final states are insensitive to the initial field strength. • For vertical fields, α~ 0.1-0.8 at saturation. • Non-axisymmetric instability of toroidal fields has the same instability criterion as the axisymmetric instability of poloidal fields (Balbus & Hawley 1992), although working instability mechanisms are different from each other (Kim et al. 2000) and the former generally yields smaller values of α at saturation.

16. Properties of MRI-Driven Turbulence • Turbulence is highly anisotropic. • Most of turbulence energy is contained in large scale modes. • At well-resolved scales, the logarithmic slopes of the power spectra are similar to the -11/3 Kolmogorov spectrum of isotropic incompressible turbulence. • <By2>/8π dominates. • For poloidal fields, • <B2>/8π = 1.2 ρ0(Lz Ω)(λcΩ) • Txy = 0.61 <B2>/8π • Rxy= 0.24 Mxy • α~ 0.2-0.8 • For toroidal fields, • <B2>/8π = 0.012 ρ0(Ly Ω)(λcΩ) • Txy = 0.51 <B2>/8π • Rxy = 0.28 Mxy • α~ 0.02-0.07 (Hawley et al. 1995,1996)

17. Effects of… • Density stratification • Brandenburg et al. (1995); Stone et al. (1996); Matsuzaki et al. (1997); Miller & Stone (2000); Kim et al. (2003) • as long as λc < H, the effect of stratification is weak. • buoyancy leads to the formation of strongly magnetized corona (Pgas << Pmag) • Radiation • Blaes & Socrates (2001); Turner et al. (2002) • radiation pressure reduces the growth rates of MRI, but does not change the instability criterion. • photon diffusion maintains nearly isothermal conditions while creating over-dense small clumps of thermally-supported gas.

18. Effects of Non-ideal MHD ve=v + (ve – vi) + (vi-v) • Neutral-ion: ambipolar diffusion • Bales & Balbus (1994); Hawley & Stone (1998) • If the coupling between ions and neutrals is weak, turbulence in ions driven by MRI does not affect the motions of neutral. • negligible if f >10-13. • Electron-neutral: Ohmic dissipation • Sano et al. (1998); Fleming et al. (2000) • Stabilizes small scale modes and weakens MHD turbulence. • No MRI-driven turbulence if Rem=cs2/ηΩ < 104 (with vanishing <B>) < 102 (with mean <B>≠0). • Electron-ion: Hall effect • Wardle (1999); Balbus & Terquem (2001); Sano & Stone (2002) • Growth rates depend on the sign of (Ω∙k)(B∙k).

19. Global Simulations 2D axisymmetric, thick disk model (Stone & Pringle 2001) • Begins from a torus and poloidal fields. • Turbulence is driven by MRI, not by convection. • Due to the mass outflow associated with turbulent eddies, the net mass accretion rate is small even when α~ 0.1-0.2. • Axisymmetric models do not maintain poloidal fields indefinitely, over-emphasize streaming modes, and do not allow toroidal field instabilities.

20. Global Simulations 3D, thick disk model (Hawley & Balbus 2002) • Accretion is driven by turbulent stresses generated by MRI. • Initially constant l  near Keplerian (l~R1/2) flows. • Midplane temperature profile T ~ R-3/2  (cf. T ~ R-3/4 in thin disks). • No significant difference between simulations with and without resistive heating.

21. 3D Global Model • The resulting flow has three well-defined components; disk / corona / jet • Inside 10 RG, the disks becomes very hot, more toroidal, and highly intermittent, forming and collapsing over the course of the simulation.

22. MRI in Protostellar Disks • Region A (R<RA) is dominated by the interaction of the stellar magnetosphere and the disk. • Region B (R< 0.1 – 1 AU) has high enoughfto sustain MHD turbulence. • Region C may experience “layered accretion” (Gammie 1996). Dead zones may cause unstable accretion once they are heated up to 1000 K (FU Orionis objects?). (Stone et al. 2000)

23. MRI in Galactic Disks • Galactic disks differ from conventional accretion disks in that the former are self-gravitating, have flat rotation curves, and are vertically more compressed. • ISM in galaxies including our own is turbulent. • σ ~ 7 (CNM), 11 (WNM)  km s-1 in the Milky Way(Heiles & Troland 2003) • σz ~ 6-10 km s-1  for extended face-on galaxies(Dickey et al. 1990) • Sellwood & Balbus (1999) proposed that MRI may operate in galactic disks as well, responsible for the observed turbulence. • Kim, Ostriker, & Stone (2003)’s 3D simulations found • B ~ 2μG • σx ~ σy ~ 4 km s-1, σz ~ 2 km s-1 • MRI-driven turbulence interacts with self-gravity to form self-gravitating clouds of mass M=a few 107 M.

24. Summary • MRI is a very powerful mechanism to turn free energy sources (e.g., angular velocity and thermal energy) into turbulence energy in astrophysical disks. • While its importance to galactic turbulence remains to be seen, MRI is responsible for angular momentum transport in accretion disks.