Earth-like planets orbiting M-dwarfs are prominent future targets when searching for life outside the solar system. We apply our newly developed Coupled Atmosphere Biogeochemistry model to investigate the coupling between the biosphere, geosphere and atmosphere to gain deeper insight into the atmospheric evolution of Earth-like planets orbiting M-dwarfs. Our main goal is to understand better atmospheric processes affecting biosignatures and climate on such worlds. Furthermore, this is the first study to our knowledge which applies an automated chemical pathway analysis quantifying the production and destruction pathways of O$_2$ for an Earth-like planet with an Archean O$_2$ abundance orbiting in the habitable zone of the M-dwarf AD Leo. Results suggest that the main production arises in the upper atmosphere from CO$_2$ photolysis followed by catalytic HO$_x$ reactions. The strongest destruction does not take place in the troposphere, as was the case in Gebauer et al. (2017) for an early-Earth analog planet around the Sun, but instead in the middle atmosphere where H$_2$O photolysis is the strongest. This result was driven by the strong Lyman-$alpha$-radiation output of AD Leo, which efficiently photolyzes H$_2$O. Results further suggest that early Earth-like atmospheres of planets orbiting an M-dwarf like AD Leo are in absolute terms less destructive for atmospheric O$_2$ than for early-Earth analog planets around the Sun despite higher concentrations of reduced gases such as e.g. H$_2$, CH$_4$ and CO. Hence the net primary productivity (NPP) required to produce the same amount of atmospheric O$_2$ at the surface is reduced. This implies that a possible Great Oxidation event, analogous to that on Earth, would have occurred earlier in time in analog atmospheres around M-dwarfs.
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