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 3-D 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 surface layers. The fastest growth rate observed is five times the orbital frequency and occurs on nearly-vertical magnetic fields. The spectrum of linear modes agrees with a WKB analysis indicating still faster growth at unresolved scales, with a maximum proportional to the gravity and inversely proportional to the gas sound speed. Disturbances reaching non-linear amplitudes steepen into trains of shocks similar to a 1-D periodic non-linear 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 field, and the structure is disrupted when the ram pressure exceeds the magnetic pressure. The largest horizontal density variations are similar to the ratio of magnetic to gas pressure, and in our calculations are more than 100. Under the conditions considered, radiation diffuses through the inhomogeneneous flow 5x faster than through the initial hydrostatic equilibrium, and the net cooling rate is several times greater than in a similar calculation without magnetic fields showing the effects of convection. These results indicate photon bubbles may be important in cooling radiation-dominated disks.
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