Understanding plasma initiation in vacuum arc discharges can help to bridge the gap between nano-scale triggering phenomena and the macroscopic surface damage caused by vacuum arcs. We present a new twodimensional particle-in-cell tool to simulate plasma initiation in direct-current (DC) copper vacuum arc discharges starting from a single, strong field emitter at the cathode. Our simulations describe in detail how a sub-micron field emission site can evolve to a macroscopic vacuum arc discharge, and provide a possible explanation for why and how cathode spots can spread on the cathode surface. Furthermore, the model provides us with a prediction for the current and voltage characteristics, as well as for properties of the plasma like densities, fluxes and electric potentials in a simple DC discharge case, which are in agreement with the known experimental values.