Onset of nonlinear saturation for Rayleigh-Taylor growth in the presence of a full spectrum of modes

It is generally recognized that a single Rayleigh-Taylor unstable mode grows exponentially, proportional to the initial amplitude, until the amplitude is about (1/10 to (1/5 of the wavelength. The growth then becomes nonlinear, and the mode evolves into spikes and bubbles. This paper considers how this picture of the transition to nonlinearity changes when, instead of there being a single mode, there is a full spectrum of modes. We argue that nonlinear behavior begins whenever the sum of modes over a specified small region of k space becomes comparable to the wavelength. In the case of a single mode, this reduces to the usual comparison of the mode's amplitude with its wavelength \ensuremath{\lambda}. But if the modal amplitudes are smooth functions of k, the modes begin to saturate when their amplitude is comparable to ${\ensuremath{\lambda}}^{2}$/R times a dimensionless scale factor; here, R is the radius in spherical geometry, or the length of the surface in planar geometry. Given this new notion of the amplitude at which nonlinear saturation begins, we construct a simple model to estimate the net perturbation resulting from a broadband initial spectrum. We assume that modes grow exponentially until saturation occurs, and then the growth of the individual modes becomes linear in time. The model predictions in two and three dimensions are compared with Read and Young's experiments [Atomic Weapons Research Establishment Report No. 011/83, Aldermasten, 1983 (unpublished)], and to Youngs's calculations [Physica 12D, 32 (1984)]. The experimental results are used to set the single parameter characterizing the onset of nonlinearity. The model provides a complete description of a weak dependence on initial amplitude. The model can be easily extended to any situation for which one can estimate single-mode growths; results are presented regarding effects on multimode growth of spherical geometry, ablation stabilization, and interface coupling.