Surface liquid water is essential for standard planetary habitability. Calculations of atmospheric circulation on tidally locked planets around M stars suggest that this peculiar orbital configuration lends itself to the trapping of large amounts of water in kilometers-thick ice on the night side, potentially removing all liquid water from the day side where photosynthesis is possible. We study this problem using a global climate model including coupled atmosphere, ocean, land, and sea-ice components as well as a continental ice sheet model driven by the climate model output. For a waterworld we find that surface winds transport sea ice toward the day side and the ocean carries heat toward the night side. As a result, night-side sea ice remains O(10 m) thick and night-side water trapping is insignificant. If a planet has large continents on its night side, they can grow ice sheets O(1000 m) thick if the geothermal heat flux is similar to Earths or smaller. Planets with a water complement similar to Earths would therefore experience a large decrease in sea level when plate tectonics drives their continents onto the night side, but would not experience complete day-side dessication. Only planets with a geothermal heat flux lower than Earths, much of their surface covered by continents, and a surface water reservoir O(10 %) of Earths would be susceptible to complete water trapping.
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