  The Sun is a source of high energy neutrinos (E > 10 GeV) produced by cosmic
ray interactions in the solar atmosphere. We study the impact of three-flavor
oscillations (in vacuum and in matter) on solar atmosphere neutrinos, and
calculate their observable fluxes at Earth, as well as their event rates in a
kilometer-scale detector in water or ice. We find that peculiar three-flavor
oscillation effects in matter, which can occur in the energy range probed by
solar atmosphere neutrinos, are significantly suppressed by averaging over the
production region and over the neutrino and antineutrino components. In
particular, we find that the relation between the neutrino fluxes at the Sun
and at the Earth can be approximately expressed in terms of phase-averaged
``vacuum'' oscillations, dominated by a single mixing parameter (the angle
theta_23).
