On the relation between initial conditions and late stages of Rayleigh-Taylor instabilities

Abstract

It is well known that the geometry and spacing of structures (diapirs) due to Rayleigh-Taylor instabilities developing in initially horizontal layers depend on the viscosities, the initial layer thicknesses and the boundary conditions of the layers. As all these factors have been investigated previously by other workers, this study examines systematically how the final wavelength and geometry of Rayleigh-Taylor instabilities are influenced by the shape of the initial deformable interface. Emphasis is placed on the distinction between the characteristic wavelength λ c (which is potentially the fastest growing) and the dominant wavelength λ d (which dominates the final stage): they may differ significantly. Assuming two layers of equal viscosities, equal or different thicknesses, and a rigid top and bottom, a series of two-dimensional finite element models is calculated varying the wavelength λ o and the amplitude h o of the initial perturbations. For harmonic initial perturbations, Fourier analysis of the early stages reveals the development and nonlinear “feeding” of higher orders which cannot be predicted by linear stability analysis. If these higher orders grow significantly faster than the initial wavelength they may overtake and dominate the final stage. As a result, h o , λ o -space is separated into regimes in which the characteristic wavelength dominates and in which the initial wavelength has sufficiently large amplitudes to control the final stage. In some cases the final wavelength differs from the characteristic wavelength by a factor as large as 4. Generalisation of the results for arbitrary structures is attempted using only growth rate curves known from linear stability analysis. Faulting, folding, differential erosion and deposition, and episodic sedimentation are proposed as potential candidates for initiating wavelengths different from that characteristic in the case of salt structures. In the case of a thin source layer the existence of diapiric overhangs and pendant peripheral lobes are found to depend on the initial wavelength. Fourier analysis is proposed as a powerful tool for determining the growth rates of geological structures which depart from sinusoidal geometries.