Nonlinear kink modes for supersonic vortex sheets

Both recent large-scale numerical simulations and time-dependent asymptotic nonlinear wave theories reveal the prominence of kink modes in the nonlinear instability of supersonic vortex sheets. These kink modes are nonlinear traveling waves that move along the vortex sheet at various speeds and have a wave structure consisting of a kink in the slip stream bracketed by shocks and rarefactions emanating from each side of the kink. Here an explicit construction is developed for calculating all the nonlinear kink modes that bifurcate from a given unperturbed contact discontinuity. This construction is applied at small amplitudes to provide a completely independent confirmation of the asymptotic nonlinear wave theories through a static bifurcation analysis. For the unperturbed vortex sheet, bifurcation diagrams at large amplitudes are also computed for several interesting density ratios and Mach numbers. These results are applied at large amplitude to explain some of the phenomena observed in numerical simulations.