By using weakly $\widetilde{\mathcal{D}}^+_J$ (resp. $\mathcal{D}^+_J$ )-closed technique firstly introduced by Tan, Wang, Zhou and Zhu, we will give a characterization of tamed and weakened tamed four-manifolds, and an almost Kaehler version of Nakai-Moishezon criterion for almost complex four-manifolds.