This paper proposes an effective three-dimensional compressive sensing method for the phased array weather radar (PAWR), which is capable of three-dimensional observation with spatially and temporally high resolution. Because of the large amount of observation data, which is approximately 1 gigabyte per minute, data compression is an essential technology to conduct a network observation by multiple PAWRs. Even though many conventional studies applied compressive sensing (CS) to weather radar measurements, their reconstruction quality should be further improved. To this end, we define a cost function for a three-dimensional recovery exploiting not only local similarity but also global redundancy of weather radar measurements. Since the cost function is convex, we can derive an efficient algorithm based on the standard convex optimization techniques. Simulation results show that the proposed method achieves normalized errors less than 10% for 25% compression ratio with outperforming conventional two-dimensional methods.