Weakly nonlinear incompressible Rayleigh-Taylor instability in spherical and planar geometries

Abstract

The relationship between the weakly nonlinear (WN) solutions of the Rayleigh-Taylor instability in spherical geometry [Zhang et al., Phys. Plasmas 24, 062703 (2017)] and those in planar geometry [Wang et al., Phys. Plasmas 19, 112706 (2012)] is analyzed. In the high-mode perturbation limit (Pn(cos θ), n≫1), it is found that at the equator, the contributions of mode P2n along with its neighboring modes, mode P3n along with its neighboring modes, and mode Pn at the third order along with its neighboring modes are equal to those of the second harmonic, the third harmonic, and the third-order feedback to the fundamental mode, respectively, in the planar case with a perturbation of the same wave vector and amplitude as those at the equator. The trends of WN results in spherical geometry towards the corresponding planar counterparts are found, and the convergence behaviors of the neighboring modes of Pn, P2n, and P3n are analyzed. Moreover, the spectra generated from the high-mode perturbations in the WN regime are provided. For low-mode perturbations, it is found that the fundamental modes saturate at larger amplitudes than the planar result. The geometry effect makes the bubbles at or near the equator grow faster than the bubbles in planar geometry in the WN regime.