The aim of this work is to propose a statistical model for spatio-temporal data on meshed surfaces based on the SPDE modeling approach. To do so, we consider a class of advection-diffusion SPDEs defined on smooth compact orientable closed Riemannian manifolds of dimension 2, and their discretization using a Galerkin approach. We show how this approach allows to easily propose scalable algorithms for the simulation and prediction of Gaussian random fields that are solutions to the discretized SPDE.