Nonlinear evolution of an interface in the Richtmyer-Meshkov instability

The linear theory of the Richtmyer-Meshkov instability derived by Wouchuk and Nishihara [Phys. Plasmas 4, 3761 (1997)] indicates that the instability is driven by the nonuniform velocity shear left by transmitted and reflected rippled shocks at a corrugated interface. In this work, the nonlinear evolution of the interface has been investigated as a self-interaction of a nonuniform vortex sheet with a density jump. The theory developed shows the importance of the finite density jump and the finite initial corrugation amplitude of the interface. By introducing Lagrangian markers on the interface with proper kinematic boundary conditions, it is shown that stretching and shrinking of the interface occur locally even in the tangential direction. This causes deformation of bubble and spike profiles depending on the Atwood number. The vorticity on the interface for a finite density jump is not conserved in the nonlinear regime. Our results suggest that the spiral structure of the spike is due to local increase and decrease of the vorticity on the interface. Nonlinear analysis shows that the large initial amplitude of the corrugation results in rapid increase of the vorticity, which may also explain the fast roll up motion of the spiral for large amplitudes. With the use of the asymptotic linear growth rate, the nonlinear evolution of the instability is uniquely determined from the initial corrugation amplitude of the interface, the Atwood number, and the incident shock intensity. There is no need to use an impulsive formulation. The analytical nonlinear growth agrees well with the experiment [Dimonte , Phys. Plasmas 3, 614 (1996)]. The theory reveals nonlinear properties of the instability, such as the time evolution of the interface profiles and the vorticity on the interface, and also their dependence on the Atwood number and the corrugation amplitude.