Linear Theory of the Rayleigh-Taylor Instability at a Discontinuous Surface of a Relativistic Flow

We address the linear stability of a discontinuous surface of a relativistic flow in the context of a jet that oscillates radially as it propagates. The restoring force of the oscillation is expected to drive a Rayleigh-Taylor instability (RTI) at the interface between the jet and its cocoon. We perform a linear analysis and numerical simulations of the growth of the RTI in the transverse plane to the jet flow with a uniform acceleration. In this system, an inertia force due to the uniform acceleration acts as the restoring force for the oscillation. We find that not only the difference in the inertia between the two fluids separated by the interface but also the pressure at the interface helps to drive the RTI because of a difference in the Lorenz factor across the discontinuous surface of the jet. The dispersion relation indicates that the linear growth rate of each mode becomes maximum when the Lorentz factor of the jet is much larger than that of the cocoon and the pressure at the jet interface is relativistic. By comparing the linear growth rates of the RTI in the analytical model and the numerical simulations, the validity of our analytically derived dispersion relation for the relativistic RTI is confirmed.