The diffusion of N adatoms on a Ga-rich GaN(0001) surface has been studied using density-functional theory. The configuration of Ga adatoms on a Ga-rich GaN surface has been identified. The first adlayer Ga adatoms are on top of the terminating substrate Ga atoms, and the outmost adlayer Ga adatoms exist randomly at the T4 or H3 sites. A very different diffusivity of N adatoms on a Ga-rich GaN(0001) surface has been found. The excess Ga adatoms on a GaN(0001) surface reduce the diffusion barrier by 0.75eV and influence the migration path. It seems that bilayer Ga adatoms are helpful for N atom diffusion.