The atmosphere is a thermo-hydrodynamical complex system and provides oxygen to most animal life at the Earth's surface. However, the detection of complexity for the atmosphere remains elusive and debated. Here we develop a percolation-based framework to explore its structure by using the global air temperature field. We find that the percolation threshold is much delayed compared with the prototypical percolation model and the giant cluster eventually emerges explosively. A finite-size-scaling analysis reveals that the observed transition in each atmosphere layer is genuine discontinuous. Furthermore, at the percolation threshold, we uncover that the boundary of the giant cluster is self-affine, with fractal dimension d_{f}, and can be utilized to quantify the atmospheric complexity. Specifically, our results indicate that the complexity of the atmosphere decreases superlinearly with height, i.e., the complexity is higher at the surface than at the top layer and vice versa, due to the atmospheric boundary forcings. The proposed methodology may evaluate and improve our understanding regarding the critical phenomena of the complex Earth system and can be used as a benchmark tool to test the performance of Earth system models.