Oscillatory instability of radiative shocks with transverse magnetic field: Linear analysis and nonlinear simulations

We examine the stability of plane-parallel radiative shocks with a transverse magnetic field, for radiative cooling laws A ∝ ρ²T^α. The instability of interest is a global thermal instability which drives a periodic oscillation of the shock front and corresponding oscillation in the instantaneous shock speed. The two dimensionless parameters determining the stability of the system are α, the exponent of the power law, and the Alfvén Mach number M_A ≡ v_s/v_A, where v_s is the average shock speed and v_A is the Alfvén velocity in the preshock medium. We determine the stable and unstable regions of this two-dimensional parameter space for the fundamental mode and the first seven overtone modes through linear analysis of the hydrodynamical equations. As expected, the magnetic field has a stabilizing influence. For α > 0 even a relatively weak magnetic field (M_A < 8) can stabilize against all modes. For α > 0.5 even weaker fields (M_A < 33) are sufficient to suppress thermal instability. We also determine the linear growth rates and frequencies of the fundamental mode and the first and second overtones for 12 pairs of the parameters α and M_A. A fully dynamical numerical simulation of these 12 cases confirms the results of the linear analysis and also explores the nonlinear behavior of the oscillations. In most of the unstable cases the amplitude of the shock front oscillation saturates at a level of 5%-10% of the length of the cooling region. In the least stable cases, however, the flow develops multiple shocks. Using a simple approximation to the cooling function for shocked interstellar gas, we show that v_s ≲ 160 km s^-1 radiative shocks in interstellar gas with n_H ≲ 0.4 cm^-3 (the "warm ionized medium" or "warm neutral medium") may be magnetically stabilized. In higher density gas, however, the magnetic field is not strong enough to appreciably affect the shock stability.