We study the 2D system of incompressible gravity driven Euler equations in the neighborhood of a particular smooth density profile ρ 0 (x)
), where ξ is a nonconstant solution of ξ = ξ ν+1 (1 -ξ ), L 0 > 0 is the width of the ablation region, ν > 1 is the thermal conductivity exponent, and ρ a > 0 is the maximum density of the fluid. The linearization of the equations around the stationary solution (ρ 0 , 0, p 0 ), ∇ p 0 = ρ 0 g leads to the study of the Rayleigh equation for the perturbation of the velocity at the wavenumber k:
We denote by the terms 'eigenmode and growth rate' an L 2 (R) solution of the Rayleigh equation associated with a value of γ . The purpose of this paper is twofold:
• derive the following expansion in k L 0 , for small k L 0 , of the unique reduced linear growth rate γ √ gk
where a 2 is explicitly known, provided ν > 2,
• prove the nonlinear instability result for small times in the neighborhood of a general profile ρ 0 (x) such that k 0 (x) = ρ 0 (x) ρ 0 (x) is regular enough, bounded, and k 0 (x)(ρ 0 (x)) -1 2 bounded (which is the case for ρ a ξ( x L 0
)), thanks to the existence of Λ such that γ ≤ Λ for all possible growth rates and at least one growth rate γ belongs to ( Λ 2 , Λ). This generalizes the result of Guo and Hwang [Y. Guo, H.J. Hwang, On the dynamical Rayleigh-Taylor instability, Arch. Ration. Mech. Anal. 167 (3) (2003) 235-253], which was obtained in the case ρ 0 (x) ≥ ρ l > 0.