Next generation communication networks are anticipated to make use of renewable energy sources, e.g., solar and wind power, to reduce carbon footprints and achieve an environmentally sustainable system. However, renewable energy sources have the limitation of unstable availability and capacity, which introduces new challenges for network planning and resource management. In this paper, adaptive resource management is introduced for wireless mesh networks that are powered by sustainable energy sources. The objective is to address the unreliability of the energy supply and to maximize the energy sustainability of the network, or equivalently, minimize the probability that mesh access points (APs) deplete their energy and go out of service. Specifically, the energy buffer of a mesh AP is modeled as a G/G/1 queue and a diffusion approximation is applied to analyze the transient evolution of the queue length and energy depletion duration. Based on the analysis, a resource management scheme is proposed to adaptively distribute traffic over various relay paths across the network and a distributed admission control strategy is applied to further guarantee high resource utilization under the energy sustainability constraint. By considering the first and second order statistics of the energy charging and discharging processes, it is demonstrated that the proposed scheme outperforms some existing state-of- the-art solutions.