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Global weather forecast is an important spatial-temporal prediction problem, which can provide numerous societal benefits such as extreme weather forewarning, traffic scheduling, and agricultural planning. Though many spatial-temporal prediction models have been proposed, they suffer from two drawbacks for global weather forecasts, namely 1) ignoring the physical mechanism and spherical characteristics and 2) not effectively exploiting the global and local correlations. To address the above drawbacks, in this paper, we formalize global weather state dynamics as partial differential equations (PDEs) in spherical space and infer the state of the global weather system by solving these PDEs. Specifically, we use Green’s function method to solve the PDEs and find that the solution of the spherical PDEs can be obtained by the spherical convolution. We further proposed a novel Spherical Neural Operator, SNO, which consists of spherical convolution and vanilla convolution. The former is used to solve these PDEs and model the global correlations in spherical space, and the latter is used to capture the local correlations. Upon the operator, a global weather prediction model is developed. Extensive experimental results demonstrate the effectiveness and superiority of our method over state-of-the-art approaches.