Atmospheric pressure glow discharge above water anode has attracted much attention due to its extensive potential applications, where self-organized patterns can be formed. Usually, they are time-invariant under constant conditions. In this article, some peculiar time-dependent patterns are formed, among which concentric ring patterns like single, triple, quadruple, and quintuple rings are observed in water-anode discharge system for the first time. Accompanying the increasing complexity of patterns, results indicate that water conductivity evolves with time. Optical emission spectroscopy reveals that before reaching constant values, electron density decreases, while electron and gas temperatures increase as time elapses. Interestingly, the temporal variations of electric conductivity and plasma parameters are correlated with the evolution of patterns. Based on the phenomena mentioned above, the pattern evolution is qualitatively explained based on the Turing mechanism at last. We think these results will shed light on formation mechanism of patterns in atmospheric pressure glow discharge with water anode.