A simple analytical model is presented to study hydrodynamic perturbation growth in the start-up phase in laser fusion, namely propagation of a rippled shock driven by non-uniform laser ablation induced by initial target roughness or non-uniform laser irradiation. These perturbation growths are very important because they seed the Rayleigh-Taylor (RF) instability in the subsequent acceleration and stagnation phases. We investigate temporal evolutions of the shock front and the ablation surface. In the result, we have found that the shock front ripples oscillate and decay in both cases of uniform laser irradiation on a target with a rippled surface and non-uniform laser irradiation on a smooth target surface. On the other hand, we obtain that there is the asymptotic value of the ablation surface deformation in the former case and the asymptotic growth rate of the ablation surface ripple in the latter case. Approximate formulas expressing both temporal evolutions of the shock front and the ablation surface are obtained in the weak shock limit. These formulas are also applicable in a relatively strong shock. We also investigate the case of oscillating non-uniform laser irradiation with time. The oscillation frequency dependences of the shock front ripple and the growth rate of the ablation surface ripple are discussed.
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