Bell Plesset Effects on Rayleigh Taylor Instability of Three Dimensional Spherical Geometry

TL;DR

This paper develops a multi-mode, weakly nonlinear theory for Rayleigh-Taylor instability on a spherical interface, revealing how Bell-Plesset effects significantly amplify growth and favor axisymmetric mode energy transfer, with implications for astrophysics and fusion.

Contribution

It extends prior RTI analyses to fully three-dimensional, dynamic backgrounds, incorporating mode couplings and Bell-Plesset effects, and highlights the dominance of axisymmetric modes in convergence-driven instabilities.

Findings

Bell-Plesset effects amplify RTI growth by orders of magnitude.

Nonlinear mode coupling favors energy transfer into axisymmetric modes.

Second order amplitudes remain small, validating perturbation theory.

Abstract

We develop a weakly nonlinear, multi-mode theory for the Rayleigh-Taylor instability (RTI) on a time-varying spherical interface, fully incorporating mode couplings and the Bell-Plesset (BP) effects arising from interface convergence. Our model extends prior analyses, which have been largely restricted to static backgrounds, 2D cylindrical geometries, or single-mode initial condition. We present a framework capable of evolving arbitrary, fully three-dimensional initial perturbations on a dynamic background. At the first order, mode amplitudes respond to the time-varying interface acceleration with an exponential-like growth, in qualitative agreement with classic static results. At second order, nonlinear mode coupling reveals a powerful selection rule: energy is preferentially channeled into axisymmetric (m=0) modes. We find that the BP effects dramatically amplify the instability growth by a few orders of magnitude, with this amplification being even more significant for second order couplings. Despite this strong channeling, the second order amplitudes remain small relative to the first order, validating the perturbative approach. These findings offer new physical insights into time-dependent interface instabilities relevant to applications such as astrophysical shell collapse and inertial confinement fusion, highlighting the uniquely dominant role of axisymmetric modes in BP-driven convergent flows.