The Vertical Structure of Radiation Dominated Accretion Disks Omer Blaes with Shigenobu Hirose and Julian Krolik

How is (turbulent!) dissipation distributed vertically? What role (if any) do convection and Poynting flux play in the vertical transport of energy? Are thermal (and “viscous”) instabilities in the radiation dominated regime real, or are they merely artifacts of a bad choice of stress prescription? How big are fluctuations about equilibrium? Do magnetic forces play any role in hydrostatic support? Huge Theoretical Uncertainties Have Plagued Us for Years - Even In the Standard Geometrically Thin, Optically Thick Disk Model

Hydrostatic equilibrium: Expectations in Radiation-Dominated Regime A radially constant disk half-thickness:

Radiative equilibrium: After vertical and time-averaging, this must be given by turbulent stress times rate of strain: Expectations in Radiation-Dominated Regime A vertically constant dissipation rate per unit volume: (Shakura & Sunyaev 1976)

Ad Hoc Prescription I: The Density Profile IF we choose the dissipation per unit MASS to be spatially constant as well, then the density must be constant. The vertical density profile is completely unconstrained! Convective instability! (Bisnovatyi-Kogan & Blinnikov 1977) But perhaps the dissipation per unit mass is not constant???

Ad Hoc Prescription II: The Stress/Pressure Relation The radial midplane temperature and surface density profiles are completely unconstrained unless we adopt, e.g., an alpha prescription for the vertically averaged stress. If we choose, then the disk is thermally and “viscously” unstable if. But other choices are possible, e.g. or (Sakimoto & Coroniti 1981, …, Merloni & Fabian 2002, …) Or perhaps much of the accretion power is dissipated in a corona above the disk? (Svensson & Zdziarski 1994)

No Observational Evidence for Radiation Pressure Thermal/“Viscous” Instabilities - Except Perhaps in GRS Belloni et al. (1997)

-Turner et al. (2003) MRI Turbulence Can be Highly Compressible in Radiation Dominated Regime Silk damping of turbulence may be important. (Agol & Krolik 1998)

-Turner et al. (2005) F g “Photon Bubble Instability”

Stratified Shearing Box Simulations of MRI Turbulence x (radial) y (azimuthal) z (vertical) Cartesian box corotating with fluid at center of box. Boundary conditions are shearing periodic in x, periodic in y, outflow in z.

Time Altitude 25% of magnetic energy generated in turbulence buoyantly rises and dissipates in outer layers. Does this produce a hot corona? Uncertain, as simulation was isothermal. Miller & Stone (2000)

Thermodynamically consistent, radiation MHD simulations in vertically stratified shearing boxes: PaperBlack Hole Mass R/(GM/c 2 )Thermal Pressure Turner (2004) 10 8 M200P rad >>P gas Hirose et al. (2006) 6.62 M300P rad <<P gas Krolik/Blaes et al. (2006) 6.62 M150P rad ~P gas Hirose et al. (2008, in prep.) 6.62 M30P rad >>P gas

x (radial) y (azimuthal) z (vertical) L x =0.45H, 48 zones L y =1.8H, 96 zones L z =8.4H, 896 zones The Simulation Domain (Apologies to Arthur C. Clarke and Stanley Kubrick)

Heating vs. Cooling Radiation, Gas Internal, Magnetic, and Turbulent Kinetic Energies Energy Balance - NO Thermal Instability!

Heating vs. Cooling Radiation, Gas Internal, Magnetic, and Turbulent Kinetic Energies

Time-Averaged Vertical Dissipation Profile Most of the dissipation is concentrated near midplane.

Turbulence near Midplane is Incompressible -----Silk Damping is Negligible

Time Averaged Vertical Energy Transport

Density is far from constant with height. Density profile at 200 orbits. Time-averaged density profile.

The Vertically-Averaged Stress

 r  /P tot  r  /(P tot P gas ) 1/2  r  /P gas

Time-averaged Radiation, Gas, and Magnetic Pressure Profiles

Vertical Hydrostatic Balance t = 200 orbits

Parker Instability g B

Parker is Clearly Present t = 200 orbits

Large Density Fluctuations at Effective and Scattering Photospheres -upper effective photosphere at t=200 orbits

Strong density fluctuations, at both scattering and effective photospheres. Strong fluctuations also seen at effective Photosphere in previous simulations with P rad >>P gas and P rad ~P gas. Photospheric Density Fluctuations

Overall Vertical Structure for all P rad /P gas Regimes MRI - the source of accretion power Photospheres Parker Unstable Regions Parker Unstable Regions P mag >P rad, P gas P rad, P gas >P mag

Conclusions No evidence for radiation pressure driven thermal instability, despite fact that turbulent stresses may be tracking total pressure (causal direction is OTHER way around, though!). Dissipation is concentrated near disk midplane, with no energetically significant corona. Upper layers are always supported by magnetic fields, even well beneath the photospheres. (Reflection modelers beware!) Parker instability dominates, and drives strong density fluctuations in all radiation/gas pressure regimes. Photon bubble instability is unresolved in this simulation. Spectra and color correction factors: magnetic field support should harden spectra, density fluctuations should soften spectra. Which dominates?

Caveats and Uncertainties Simulations are expensive, and much more work needs to be done to address the following issues: Numerical convergence with increased resolution and box size in all three directions (particularly radial and azimuthal). How does initial magnetic field topology affect things? (We start with a twisted azimuthal flux tube with net azimuthal flux, but no net poloidal flux.) Most of our dissipation is numerical and is captured at the grid scale. Viscous and resistive scales are therefore identical (i.e. Prandtl number is unity). Simulations in non-stratified shearing boxes show that this might matter.

More Details on the (Time-Averaged) Energy Balance Stress Dissipation Rate Divergences of Poynting flux, Gas energy advection, Radiation energy advection, and Radiative diffusion, Total of last three matches dissipation rate.

Flux from top Flux from bottom

Photosphere Evolution