  Temperate terrestrial planets orbiting low-mass stars are subject to strong
tidal forces. The effects of gravitational tides on the solid planet and that
of atmospheric thermal tides have been studied, but the direct impact of
gravitational tides on the atmosphere itself has so far been ignored. We first
develop a simplified analytic theory of tides acting on the atmosphere of a
planet. We then implement gravitational tides into a general circulation model
of a static-ocean planet in a short-period orbit around a low-mass star -- the
results agree with our analytic theory. Because atmospheric tides and
solid-body tides share a scaling with the semi-major axis, we show that there
is a maximum amplitude of the atmospheric tide that a terrestrial planet can
experience while still having a solid surface; Proxima Centauri b is the poster
child for a planet that could be geophysically Earth-like but with atmospheric
tides more than 500$\times$ stronger than Earth's. In this most extreme
scenario, we show that atmospheric tides significantly impact the planet's
meteorology -- but not its climate. Two possible modest climate impacts are
enhanced longitudinal heat transport and cooling of the lowest atmospheric
layers. The strong radiative forcing of such planets dominates over
gravitational tides, unlike moons of cold giant planets, such as Titan. We
speculate that atmospheric tides could be climatologically important on planets
where the altitude of maximal tidal forcing coincides with the altitude of
cloud formation and that the effect could be detectable for non-Earth-like
planets subject to even greater tides.
