  Water cycling between Earth's mantle and surface has previously been modelled
and extrapolated to rocky exoplanets, but these studies neglected the host
star. M-dwarf stars are more common than Sun-like stars and at least as likely
to host temperate rocky planets (M-Earths). However, M dwarfs are active
throughout their lifetimes; specifically, X-ray and extreme ultraviolet (XUV)
radiation during their early evolution can cause rapid atmospheric loss on
orbiting planets. The increasing bolometric flux reaching M-Earths leads to
warmer, moister upper atmospheres, while XUV radiation can photodissociate
water molecules and drive hydrogen and oxygen escape to space. Here, we present
a coupled model of deep-water cycling and water loss to space on M-Earths to
explore whether these planets can remain habitable despite their volatile
evolution. We use a cycling parameterization accounting for the dependence of
mantle degassing on seafloor pressure, the dependence of regassing on mantle
temperature, and the effect of water on mantle viscosity and thermal evolution.
We assume the M dwarf's XUV radiation decreases exponentially with time, and
energy-limited water loss with 30% efficiency. We explore the effects of
cycling and loss to space on planetary water inventories and water
partitioning. Planet surfaces desiccated by loss can be rehydrated, provided
there is sufficient water sequestered in the mantle to degas once loss rates
diminish at later times. For a given water loss rate, the key parameter is the
mantle overturn timescale at early times: if the mantle overturn timescale is
longer than the loss timescale, then the planet is likely to keep some of its
water.
