Explicit expressions for the evolution of single-mode Rayleigh-Taylor and Richtmyer-Meshkov instabilities at arbitrary Atwood numbers

We present explicit analytic expressions for the evolution of the bubble amplitude in Rayleigh-Taylor (RT) and Richtmyer-Meshkov RM instabilities. These expressions are valid from the linear to the nonlinear regime and for arbitrary Atwood number A. Our method is to convert from the linear to the nonlinear solution at a specific value eta* of the amplitude for which explicit analytic expressions have been given previously for A=1 K. O. Mikaelian, Phys. Rev. Lett. 80, 508 (1998). By analyzing a recent extension of Layzer's theory to arbitrary A V. N. Goncharov, Phys. Rev. Lett. 88, 134502 (2002), we find a simple transformation that generalizes our solutions to arbitrary A. We compare this model with another explicit model attributed to Fermi and with numerical simulations. Fermi's model agrees with numerical simulations for the RT case but its extension to the RM case disagrees with simulations. The model proposed here agrees with hydrocode calculations for both RT and RM instabilities.