  We use the Met Office Unified Model to explore the potential of a tidally
locked M dwarf planet, nominally Proxima Centauri b irradiated by a quiescent
version of its host star, to sustain an atmospheric ozone layer. We assume a
slab ocean surface layer, and an Earth-like atmosphere of nitrogen and oxygen
with trace amounts of ozone and water vapour. We describe ozone chemistry using
the Chapman mechanism and the hydrogen oxide (HO$_x$, describing the sum of OH
and HO$_2$) catalytic cycle. We find that Proxima Centauri radiates with
sufficient UV energy to initialize the Chapman mechanism. The result is a thin
but stable ozone layer that peaks at 0.75 parts per million at 25 km. The
quasi-stationary distribution of atmospheric ozone is determined by photolysis
driven by incoming stellar radiation and by atmospheric transport. Ozone mole
fractions are smallest in the lowest 15 km of the atmosphere at the sub-stellar
point and largest in the nightside gyres. Above 15 km the ozone distribution is
dominated by an equatorial jet stream that circumnavigates the planet. The
nightside ozone distribution is dominated by two cyclonic Rossby gyres that
result in localized ozone hotspots. On the dayside the atmospheric lifetime is
determined by the HO$_x$ catalytic cycle and deposition to the surface, with
nightside lifetimes due to chemistry much longer than timescales associated
with atmospheric transport. Surface UV values peak at the substellar point with
values of 0.01 W/m$^2$, shielded by the overlying atmospheric ozone layer but
more importantly by water vapour clouds.
