A weakly nonlinear theory for the dynamical Rayleigh–Taylor instability

The dynamics of an interface between two incompressible, inviscid, irrotational, and immiscible liquids with densities ρ1 and ρ2 under the influence of a time-dependent gravitational field g(t) is investigated. A Hamiltonian formulation of the system is adopted leading to a perturbative expansion of the equations of motion for the canonical variables. Equations, accurate up to third order in the perturbation amplitude are derived. They are able to describe the initial stage of instability “saturation.” The latter equations are integrated iteratively for two standard limiting cases: constant gravity (classical Rayleigh–Taylor instability), g(t)≡g0, and impulsive Richtmyer–Meshkov loading, g(t)=v0δ(t−t0). The comparative growth of various two-dimensional structures and rectangular and hexagonal cells is evaluated. Surface tension effects are considered.