The Most Dangerous Seed: Nonlinear Optimal Perturbations in Rayleigh-Taylor Instability

Abstract

Long-term instabilities in astrophysical fluids are inherently nonlinear, where even small-amplitude perturbations can trigger dramatic instability. However, owing to the complex interactions among non-normal modes, the perturbation structures responsible for the greatest growth remain poorly understood. In this paper, we employ the nonlinear optimization method known as the conditional nonlinear optimal perturbation (CNOP) to identify the most dangerous initial velocity perturbation, i.e., the perturbation that maximizes the kinetic energy growth in the two-dimensional compressible Rayleigh-Taylor instability in astrophysical hydrodynamics. Compared with random perturbations, the optimal perturbation forms a coherent wave-packet structure localized around the density interface. We investigate its dependence on spatial resolution and optimization time horizon through two sets of numerical experiments. For a fixed optimization time, increasing the spatial resolution produces a more sharply localized wave packet, whereas for a fixed spatial resolution, increasing the optimization time causes the wave packet to become progressively more dispersed. Furthermore, we analyze the optimal perturbations in Fourier space using the fast Fourier transform (FFT), which provides a clearer characterization of the spectral distribution. Higher-resolution simulations concentrate most of the perturbation energy into only a few dominant modes, while longer optimization times distribute the energy over a broader range of modes. These results indicate that short optimization time horizons involve relatively weak modal interactions and remain closer to the linear regime, whereas longer optimization times enhance nonlinear modal interactions, broaden the spectral distribution, and reduce the predictive capability of linear stability theory.