Analytic approach to nonlinear hydrodynamic instabilities driven by time-dependent accelerations

We extend our earlier model for Rayleigh-Taylor and Richtmyer-Meshkov instabilities to the more general class of hydrodynamic instabilities driven by a time-dependent acceleration g(t). Explicit analytic solutions for linear as well as nonlinear amplitudes are obtained for several g(t)s by solving a Schrodinger-like equation d(2)eta/dt(2)-g(t)kA eta=0, where A is the Atwood number and k is the wave number of the perturbation amplitude eta(t). In our model a simple transformation k -> k(L) and A -> A(L) connects the linear to the nonlinear amplitudes: eta(nonlinear) (k, A) similar to (1/k(L))ln eta(linear) (k(L), A(L)). The model is found to be in very good agreement with direct numerical simulations. Bubble amplitudes for a variety of accelerations are seen to scale with s defined by s = integral root g(t)dt, while spike amplitudes prefer scaling with displacement Delta x=integral[integral g(t)dt]dt.