We study the emergency communication problem in the post-disaster scenario. The emergence of Renewable Energy-enabled Based Station (REBS), which has pre-equipped energy harvesting devices, provides a new perspective to solve this problem, since the post-disaster communication scenario happens to be a Wireless Mesh Network (WMN) constituted by REBSs. However, one needs to address the unstable and inadequate power supply, insufficient capacities of the communication links, and the time-varying traffic demands accordingly during a period of time. In this paper, we deal with these problems and focus on optimizing data traffic throughput with the lowest weighted energy consumption based on the expectation of traffic demands. We firstly analyze the post-disaster communication issue and formulate this problem into a Mixed Integer Linear Programming (MILP) problem, and propose an off-line energy efficient scheme using the expectation of traffic demands. Furthermore, an on-line scheme is put forward which dynamically adjusts the off-line result with the knowledge of the current real data traffic demands. The efficiency of our proposal is demonstrated by theoretical analyses and numerical results.