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In this research, a weakly nonlinear (WN) model for the incompressible Rayleigh-Taylor instability in cylindrical geometry [Wang et al., Phys. Plasmas 20, 042708 (2013)] is generalized to spherical geometry. The evolution of the interface with an initial small-amplitude single-mode perturbation in the form of Legendre mode (Pn) is analysed with the third-order WN solutions. The transition of the small-amplitude perturbed spherical interface to the bubble-and-spike structure can be observed by our model. For single-mode perturbation Pn, besides the generation of P2n and P3n, which are similar to the second and third harmonics in planar and cylindrical geometries, many other modes in the range of P0–P3n are generated by mode-coupling effects up to the third order. With the same initial amplitude, the bubbles at the pole grow faster than those at the equator in the WN regime. Furthermore, it is found that the behavior of the bubbles at the pole is similar to that of three-dimensional axisymmetric bubbles, while the behavior of the bubbles at the equator is similar to that of two-dimensional bubbles.

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Nonlinear amplitude saturation (NAS) of the fundamental mode of Rayleigh–Taylor instability (RTI) in a finite-thickness incompressible fluid layer is investigated analytically by considering high-order corrections (HOCs) up to the ninth order. The results of classical RTI [Liu et al., Phys. Plasmas 19, 042705 (2012)] can be recovered for the normalized fluid thickness kd→∞. It is found that the NAS of the fundamental mode on the lower and upper interfaces is clearly larger than its third-order counterpart [Wang et al., Phys. Plasmas 21, 122710 (2014)] when the HOCs are considered, especially for the lower (linearly unstable) interface. Furthermore, the NAS on both interfaces exhibits the trend of convergence with increasing order of corrections.