Computations of space charge limited currents from thermionic emitters often rely on simple one-dimensional analytical models. The geometric limitations of these models are circumvented by applying the models locally over the emitter surface. The validity of the analytical models is compromised by the presence of surface roughness that intrinsically violates one-dimensionality. In this paper we will calculate the self-consistent potential distribution and emission current of a bumpy surface. It will be shown that, although surface roughness influences the emission function in the temperature limit, a space charge barrier causes smoothening in the current density so that a rough emitter, operated in space charge limited mode has an emission function that is very similar to a perfect planar emission function, given bumps smaller than the dimension of the space charge barrier.