We have investigated the effect of the surface roughness of a dielectric electrode for dielectric barrier discharge (DBD). We prepared three alumina plates made from the same material with different surface roughnesses for use as electrodes. During the repeated breakdown from the first breakdown to the stationary state, we observed gradual increases in the secondary ionization coefficient and the charge accumulated on the alumina electrodes under the application of a sinusoidal voltage in argon up to a pressure of 105 Torr. A strong correlation was also observed between them, similarly to in a previous paper on CaO film electrodes. The results suggest that increasing the surface roughness of an alumina electrode results in a larger secondary ionization coefficient and greater charge accumulation on the electrode. Moreover, it is concluded that the charges that accumulate on the alumina can be liberated from the surface as initial electrons when it acts as a cathode, depending on the polarity of the alternating gap voltage. These electrons induce the formation of an atmospheric-pressure Townsend discharge. Finally, we discuss the effect of the secondary ionization coefficient in DBD.