  Context. Emission lines formed in the transition region and corona show
dominantly redshifts and blueshifts, respectively. Aims. We investigate the
Doppler shifts in a 3D radiation MHD model of the quiet Sun and compare these
to observed properties. We concentrate on Si IV 1394 A originating in the
transition region and examine the Doppler shifts of several other spectral
lines at different formation temperatures. Methods. We construct a radiation
MHD model extending from the upper convection zone to the lower corona using
the MURaM code. In this quiet Sun model the magnetic field is self-consistently
maintained by the action of a small-scale dynamo. We synthesize the profiles of
several optically thin emission lines, formed at temperatures from the
transition region into the corona. We investigate the spatial structure and
coverage of red- and blueshifts and how this changes with line-formation
temperature. Results. The model successfully reproduces the observed change of
average net Doppler shifts from red- to blueshifted from the transition region
into the corona. In particular, the model shows a clear imbalance of area
coverage of red- vs. blueshifts in the transition region of ca. 80% to 20%. We
determine that (at least) four processes generate the systematic Doppler shifts
in our model, including pressure enhancement in the transition region,
transition region brightenings unrelated to coronal emission, boundaries
between cold and hot plasma, and siphon-type flows. Conclusions. We show that
there is not a single process that is responsible for the observed net Doppler
shifts in the transition region and corona. Because current 3D MHD models do
not yet fully capture the evolution of spicules, one of the key ingredients of
the chromosphere, most probably these have still to be added to the list of
processes responsible for the persistent Doppler shifts.
