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Rayleigh-Taylor Instability in a Relativistic Fireball on a Moving Computational Grid

We numerically calculate the growth and saturation of the Rayleigh-Taylor instability caused by the deceleration of relativistic outflows with Lorentz factor Γ = 10, 30, and 100. The instability generates turbulence whose scale exhibits strong dependence on Lorentz factor, as only modes with angular size smaller than 1/Γ can grow. We develop a simple diagnostic to measure the kinetic energy in turbulent fluctuations, and calculate a ratio of turbulent kinetic energy to thermal energy of .03 in the region affected by the instability. Although our numerical calculation does not include magnetic fields, we argue that small scale turbulent dynamo amplifies magnetic fields to nearly this same fraction, giving a ratio of magnetic to thermal energy of ~ .01, to within a factor of two. The instability completely disrupts the contact discontinuity between the ejecta and the swept up circumburst medium. The reverse shock is stable, but is impacted by the Rayleigh-Taylor instability, which strengthens the reverse shock and pushes it away from the forward shock. The forward shock front is unaffected by the instability, but Rayleigh-Taylor fingers can penetrate of order 10% of the way into the energetic region behind the shock during the two-shock phase of the explosion. We calculate afterglow emission from the explosion and find the reverse shock emission peaks at a later time due to its reduced Lorentz factor and modified density and pressure at the shock front. These calculations are performed using a novel numerical technique that includes a moving computational grid. The moving grid is essential as it maintains contact discontinuities to high precision and can easily evolve flows with extremely large Lorentz factors.