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.
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