The effects of photon bubble instability in radiation-dominated accretion disks

We examine the effects of photon bubble instability in radiation-dominated accretion disks such as those found around black holes in active galactic nuclei and X-ray binary star systems. Two- and three-dimensional numerical radiation MHD calculations of small patches of disk are used. Modes with wavelengths shorter than the gas pressure scale height grow faster than the orbital frequency in the disk surface layers. The fastest growth rate observed is 5 times the orbital frequency and occurs on nearly vertical magnetic fields. The spectrum of linear modes is in good agreement with a WKB analysis that indicates still faster growth at unresolved scales, with a maximum growth rate proportional to the gravitational acceleration and inversely proportional to the gas sound speed. Disturbances reaching nonlinear amplitudes steepen into trains of shocks similar to a one-dimensional periodic nonlinear analytic solution. Variations in propagation speed result in merging of adjacent fronts, and over time the shock spacing and amplitude increase. Growth is limited by the strength of the magnetic field. The shock train structure is disrupted when the ram pressure of the disturbances exceeds the magnetic pressure. The maximum horizontal density variations are comparable to the ratio of magnetic to gas pressure and in our calculations exceed 100. Under the conditions considered, radiation diffuses through the inhomogeneneous flow 5 times faster than through the initial hydrostatic equilibrium, and the net cooling rate is several times greater than in a similar calculation without magnetic fields that shows the effects of convection. These results indicate that photon bubbles may be important in cooling radiation-dominated accretion disks. The Shaviv type I global instability grows faster than the orbital frequency in calculations of the disk surface layers with lower boundaries of fixed temperature, but is weak or absent in calculations spanning the disk thickness.