We perform a new, detailed calculation of the flux and energy spectrum of Earth-emerging $tau$-leptons generated from the interactions of tau neutrinos and antineutrinos in the Earth. A layered model of the Earth is used to describe the variable density profile of the Earth. Different assumptions regarding the neutrino charged- and neutral-current cross sections as well as the $tau$-lepton energy loss models are used to quantify the systematic uncertainty from these on the results. A baseline simulation is then used to generate the optical Cherenkov signal from upward-moving extensive air showers generated by the $tau$-lepton decay in the atmosphere, applicable to a range of space-based instruments. We use this simulation to determine the neutrino sensitivity for $E_ u>$ 10 PeV for a space-based experiment with performance similar to that for the Probe of Extreme MultiMessenger Astrophysics (POEMMA) mission currently under study.
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