Self-consistent stability analysis of ablation fronts with small Froude numbers

The linear growth rate of the Rayleigh–Taylor instability is calculated for accelerated ablation fronts with small Froude numbers (Fr≪1). The derivation is carried out self-consistently by including the effects of finite thermal conductivity (κ∼Tν) and density gradient scale length (L). It is shown that long-wavelength modes with wave numbers kL0≪1 [L0=νν/(ν+1)ν+1 min(L)] have a growth rate γ≂√ATkg−βkVa, where Va is the ablation velocity, g is the acceleration, AT=1+O[(kL0)1/ν], and 1<β(ν)<2. Short-wavelength modes are stabilized by ablative convection, finite density gradient, and thermal smoothing. The growth rate is γ=√αg/L0+c20k4L20V2a−c0k2L0Va for 1≪kL0≪Fr−1/3, and γ=c1g/(Vak2L20)−c2kVa for the wave numbers near the cutoff kc. The parameters α and c0−2 mainly depend on the power index ν; and the cutoff kc of the unstable spectrum occurs for kcL0∼Fr−1/3≫1. Furthermore, an asymptotic formula reproducing the growth rate at small and large Froude numbers is derived and compared with numerical results.