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

TL;DR

This work develops a weakly nonlinear, multi-mode theory for Rayleigh–Taylor instability on a time-dependent spherical interface, fully incorporating Bell–Plesset convergence effects and mode coupling. By expanding perturbations in spherical harmonics and solving up to second order, it reveals a strong selection rule that concentrates nonlinear energy into axisymmetric ($m=0$) modes, with Bell–Plesset curvature dramatically amplifying growth compared with static geometries. The study demonstrates that even modest compression can amplify instability amplitudes by about two orders of magnitude, and that BP effects disproportionately enhance higher-order couplings, amplifying axisymmetric pathways. The framework informs RTI in both astrophysical shell collapse and inertial confinement fusion and highlights axisymmetric perturbations as a dominant conduit for BP-driven convergent-flow instabilities.

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 conditions. 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.

Figures (13)

• Figure 1: Demonstration of the implosion prescription: the blue curve shows $R/R_f$ as a function of $\gamma t$ (left axis), while the red curve shows $\dot R/(\gamma R_f)$ vs. $\gamma t$ (right axis).
• Figure 2: Time evolution of $\gamma^2 z_l(t)$ for modes $l = 1, 5, 10$ (solid lines), and the scaled acceleration $-\gamma^2\,\ddot R / R_f$ (dashed line). The similar shapes highlight the tight relation between $z_l$ and the interface acceleration.
• Figure 3: Heatmap of $g_l(t)$ for $l = 1$ to $20$ evolving from $\gamma t=5$ to $10$. Redder regions denote large amplitudes, while blue regions denote small amplitudes.
• Figure 4: Evolution of the growth factor $g_l(t)$ for selected modes ($l = 1, 2, 5, 10, 20$) plotted on a logarithmic scale. The curves illustrate a period of accelerated growth followed by a saturation into a linear growth regime at late times as the interface deceleration ceases.
• Figure 5: Interface shape at $\gamma t = 10$ for four initial conditions: $(l,m) = \text{(a) }(3,\pm1),\,\text{(b) }(4,0),\,\text{(c) }(4,\pm1),\,\text{(d) }(4,\pm4)$. Colors represent the normalized displacement amplitude ($\eta/R_f$), where red indicates outward displacement ($\eta > 0$) and blue indicates inward displacement ($\eta < 0$). The directions of coordinates are marked by the arrows on the bottom left.