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 Schr\"odinger-like equation d²η/dt²-g(t)kAη=0, where A is the Atwood number and k is the wave number of the perturbation amplitude η(t). In our model a simple transformation k→kL and A→AL connects the linear to the nonlinear amplitudes: ηⁿᵒⁿˡⁱⁿᵉᵃʳ(k,A)~(1/kL)ln ηˡⁱⁿᵉᵃʳ(kL,AL). 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=∫√g(t)dt, while spike amplitudes prefer scaling with displacement Δx=∫[∫g(t)dt]dt.