The Drag Instability in a 2D Isothermal C-shock

We extend the linear analysis of the drag instability in a 1D perpendicular isothermal C-shock by Gu & Chen to 2D perpendicular and oblique C-shocks in the typical environment of star-forming clouds. Simplified dispersion relations are derived for the unstable modes. We find that the mode property of the drag instability generally depends on the ratio of the transverse (normal to the shock flow) to longitudinal (along the shock flow) wavenumber. For the transversely large-scale mode, the growth rate and wave frequency of the drag instability in a 2D shock resemble those in a 1D shock. For the transversely small-scale mode, the drag instability is characterized by an unstable mode coupled with an acoustic mode primarily along the transverse direction. When the shock is perpendicular or less oblique, there exists a slowly propagating mode, which can potentially grow into a nonlinear regime and contribute to the maximum growth of the instability. In contrast, when the shock is more oblique, this slowly propagating unstable mode disappears, and the maximum growth of the drag instability is likely contributed from the transversely large-scale mode (i.e., almost 1D mode). In all cases that we consider, the magnitude of the density perturbations is significantly larger than that of the velocity and magnetic field perturbations, implying that the density enhancement governs the dynamics in the linear regime of the instability. A few issues in the linear analysis, as well as the possible astrophysical implications, are also briefly discussed.