How does salinity shape ocean circulation and ice geometry on Enceladus and other icy satellites?

Of profound astrobiological interest is that not only does Enceladus have a water ocean, but it also appears to be salty, important for its likely habitability. Here, we investigate how salinity affects ocean dynamics and equilibrium ice shell geometry and use knowledge of ice shell geometry and tidal heating rates to help constrain ocean salinity. We show that the vertical overturning circulation of the ocean, driven from above by melting and freezing and the temperature dependence of the freezing point of water on pressure, has opposing signs at very low and very high salinities. In both cases, heat and freshwater converges toward the equator, where the ice is thick, acting to homogenize thickness variations. In order to maintain observed ice thickness variations, ocean heat convergence should not overwhelm heat loss rates through the equatorial ice sheet. This can only happen when the oceans salinity has intermediate values, order $20$~psu. In this case polar-sinking driven by meridional temperature variations is largely canceled by equatorial-sinking circulation driven by salinity variations and a consistent ocean circulation, ice shell geometry and tidal heating rate can be achieved.