A two dimensional numerical study on a daytime sea-breeze front was carried out in order to investigate the interaction of the sea-breeze front with Benard-Rayleigh-type convective cells within the well-mixed boundary layer ahead of the sea-breeze front. By using a nonhydrostatic compressible dry model with a very fine spatial resolution, it was found that either a frontogenesis (FG), or a frontolysis (FL) phenomenon, occurs alternately at the foremost part of the sea-breeze head. Through these interactions, the propagation speed of the sea-breeze front varied discontinuously. The frontal structure such as the shape, vertical velocity, temperature field and so on showed periodic variations as it was affected by upward and downward motions associated with the prefrontal convective cells. Based on horizontal distribution of the vertical velocity, the periodic variation was classified into three stages: FG, transition, and FL stages. For each stage, the detailed processes of FG and FL phenomena were examined by estimating each term of the frontogenesis equation. It is found that the confluence, and tilting terms, play an important role in the FG and FL processes.