Emergence of unstable modes for classical shock waves in isothermal ideal MHD

This note studies classical magnetohydrodynamic shock waves in an inviscid fluidic plasma that is assumed to be a perfect conductor of heat as well as of electricity. For this mathematically prototypical material, it identifies, mainly numerically, two critical manifolds in parameter space, across which slow resp. fast MHD shock waves undergo emergence of a complex conjugate pair of unstable transverse modes. For slow shocks, this emergence occurs in a particularly interesting way already in the parallel case, in which it happens at the spectral value λˆ≡λ∕|ω|=0 and the critical manifold possesses a simple explicit algebraic representation. Results of refined numerical treatment show that within the set of non-parallel slow shocks the unstable mode pair emerges from two generically different spectral values λˆ=±iγ. For fast shocks, the critical manifold does not intersect the parallel regime and the emergence within the set of non-parallel fast shocks again starts from two generically different spectral values.