The linear and non linear Rayleigh-Taylor instability for the quasi isobaric profile

We study the stability of the system of the Euler equations in the neighborhood of the stationary solution associated with the quasi isobaric profile in a gravity field. This situation corresponds to a Rayleigh-Taylor type problem with a smooth base density profile which goes from 0 to ρa (of Atwood number A = 1) given by the ablation front model with a thermal conductivity exponent ν > 1. This linear analysis leads to the study of the Rayleigh equation for the perturbation of the velocity at the frequency k: − d dx (ρ0(x) du dx ) + k[ρ0(x)− g γ2 ρ ′ 0(x)]u = 0. We denote by the terms ’eigenmode and eigenvalue’ a L solution of the Rayleigh equation associated with a value of γ. Let L0 > 0 be given. The quasi isobaric profile is ρ0(x) = ρaξ( x L0 ), where ξ̇ = ξ(1−ξ). We prove that there exists Lm(k), such that, for all 0 < L0 ≤ Lm, there exists an eigenmode u such that the unique associated eigenvalue γ is in [α1, α2], α1 > 0. Its limit when L0 goes to zero is √ gk. We obtain an expansion of γ in terms of L0 as follows: