The Rayleigh-Taylor instability in a self-gravitating two-layer fluid sphere

Abstract

The Rayleigh-Taylor instability is studied in a self-gravitating two-layer fluid sphere: an inner sphere and an outer layer. The density and the viscosity are assumed to be constant in each region. Analytic expressions of the dispersion relations are obtained in inviscid and viscid cases. This examina- tion aims at the investigation of the Earth's core formation. The fluid sphere corresponds to the proto-Earth in the accretion stage. The instability is examined without rotation of the fluid sphere, while the proto-Earth is rotating. However, it is shown that the Coriolis force does not influence the conclusion in the Earth's core formation problem. The main properties of the instability are as follows: For l = 1 (where l is the subscript of a spherical harmonic Y)l m, the growth rate Σ is determined mainly by deformation of the outer layer, while, for l ⩾ 2, by deformation of the more viscous region of the inner sphere and the outer layer. The time scale of the instability is governed by the free-fall time in the case of weak viscosity, and by the viscous-diffusion time in the case of appreciable viscosity. These results are applied to the Earth's core formation problem in another paper (Ida et al., 1987), where we concluded that the Earth's core has formed through the translational mode of the Rayleigh-Taylor instability with the time-scale of 10 h.