  Tau neutrinos are expected to comprise roughly one third of both the
astrophysical and cosmogenic neutrino flux, but currently the flavor ratio is
poorly constrained and the expected flux at energies above $10^{17}$ eV is low.
We present a detector concept aimed at measuring the diffuse flux of tau
neutrinos in this energy range via a high-elevation mountaintop detector using
the radio technique. The detector searches for radio signals from upgoing air
showers generated by Earth-skimming tau neutrinos. Signals from several
antennas in a compact array are coherently summed at the trigger level,
permitting not only directional masking of anthropogenic backgrounds, but also
a low trigger threshold. This design takes advantage of both the large viewing
area available at high-elevation sites and the nearly full duty cycle available
to radio instruments. We present trade studies that consider the station
elevation, frequency band, number of antennas in the array, and the trigger
threshold to develop a highly efficient station design. Such a mountaintop
detector can achieve a factor of ten improvement in acceptance over existing
instruments with 100 independent stations. With 1000 stations and three years
of observation, it can achieve a sensitivity to an integrated
$\mathcal{E}^{-2}$ flux of $<10^{-9}$ GeV cm$^{-2}$ sr$^{-1}$ s$^{-1}$, in the
range of the expected flux of all-flavor cosmogenic neutrinos assuming a pure
iron cosmic-ray composition.
