  We use numerical simulations to study the dynamics of surface discharges,
which are common in high-voltage engineering. We simulate positive streamer
discharges that propagate towards a dielectric surface, attach to it, and then
propagate over the surface. The simulations are performed in air with a
two-dimensional plasma fluid model, in which a flat dielectric is placed
between two plate electrodes. Electrostatic attraction is the main mechanism
that causes streamers to grow towards the dielectric. Due to the net charge in
the streamer head, the dielectric gets polarized, and the electric field
between the streamer and the dielectric is increased. Compared to streamers in
bulk gas, surface streamers have a smaller radius, a higher electric field, a
higher electron density, and higher propagation velocity. A higher applied
voltage leads to faster inception and faster propagation of the surface
discharge. A higher dielectric permittivity leads to more rapid attachment of
the streamer to the surface and a thinner surface streamer. Secondary emission
coefficients are shown to play a modest role, which is due to relatively strong
photoionization in air. In the simulations, a high electric field is present
between the positive streamers and the dielectric surface. We show that the
magnitude and decay of this field are affected by the positive ion mobility.
