  We show that magnetized, radiation dominated atmospheres can support steady
state patterns of density inhomogeneity that enable them to radiate at far
above the Eddington limit, without suffering mass loss. The inhomogeneities
consist of periodic shock fronts bounding narrow, high-density regions,
interspersed with much broader regions of low density. The radiation flux
avoids the regions of high density, which are therefore weighed down by
gravity, while gas in the low-density regions is slammed upward into the shock
fronts by radiation force. As the wave pattern moves through the atmosphere,
each parcel of matter alternately experiences upward and downward forces, which
balance on average. Magnetic tension shares the competing forces between
regions of different densities, preventing the atmosphere from blowing apart.
We calculate the density structure and phase speed of the wave pattern, and
relate these to the wavelength, the density contrast, and the factor by which
the net radiation flux exceeds the Eddington limit. In principle, this factor
can be as large as the ratio of magnetic pressure to mean gas pressure, or the
ratio of radiation pressure to gas pressure, whichever is smaller. Although the
magnetic pressure must be large compared to the mean gas pressure in order to
support a large density contrast, it need not be large compared to the
radiation pressure. These highly inhomogeneous flows could represent the
nonlinear development of the "photon bubble" instability discovered by Gammie.
We briefly discuss the applicability of these solutions to astrophysical
systems.
