  We study negative streamers in ambient air using a 2D axisymmetric fluid
model. Depending on the background electric field, we observe accelerating,
steady and fading negative streamers. Fading occurs in low background fields,
when negative streamers lose their field enhancement and when their velocities
become comparable to their maximal electron drift velocities. Our focus is on
the steady propagation mode, during which streamer properties like radius and
velocity hardly change. However, this mode is unstable, in the sense that a
small change in conditions leads to acceleration or deceleration. We observe
steady negative streamers in background fields ranging from 9.19 kV/cm to 15.75
kV/cm, indicating that there is no unique steady propagation field (or
stability field). Another finding is that steady negative streamers are able to
keep propagating over tens of centimeters, with only a finite conductive length
behind their heads, similar to steady positive streamers. Approximately linear
relationships are observed between the optical diameter and properties like the
streamer velocity and the streamer head potential. From these linear relations,
we obtain rough lower bounds of about 0.27 mm to 0.35 mm for the minimal
optical diameter of steady negative streamers. The lowest background field in
which a steady negative streamer could be obtained is 9.19 kV/cm. In contrast,
steady positive streamers have recently been obtained in a background field as
low as 4.05 kV/cm. We find that the properties of steady negative and positive
streamers differ significantly. For example, for steady negative streamers the
ratio between streamer velocity and maximal electron drift velocity ranges from
about 2 to 4.5, whereas for steady positive streamers this ratio ranges from
about 0.05 to 0.26.
