Nonlinear photon bubbles driven by buoyancy

We derive an analytic model for nonlinear "photon bubble'' wave trains driven by buoyancy forces in magnetized, radiation pressure-dominated atmospheres. Continuous, periodic wave solutions exist when radiative diffusion is slow compared to the dynamical timescale of the atmosphere. We identify these waves with the saturation of a linear instability discovered by Arons; therefore, these wave trains should develop spontaneously. The buoyancy-driven waves are physically distinct from a second family of photon bubbles discovered by Gammie, which evolve into trains of gas pressure-dominated shocks as they become nonlinear. Like the gas pressure-driven shock trains, buoyancy-driven photon bubbles can exhibit very large density contrasts, which greatly enhance the flow of radiation through the atmosphere. However, steady state solutions for buoyancy-driven photon bubbles exist only when an extra source of radiation is added to the energy equation, in the form of a flux divergence. We argue that this term is required to compensate for the radiation flux lost via the bubbles, which increases with height. We speculate that an atmosphere subject to buoyancy-driven photon bubbles, but lacking this compensating energy source, would lose pressure support and collapse on a timescale much shorter than the radiative diffusion time in the equivalent homogeneous atmosphere.