  Cloud cover at the planetary limb of water-rich Earth-like planets is likely
to weaken chemical signatures in transmission spectra, impeding attempts to
characterize these atmospheres. However, based on observations of Earth and
solar system worlds, exoplanets with atmospheres should have both short-term
weather and long-term climate variability, implying that cloud cover may be
less during some observing periods. We identify and describe a mechanism
driving periodic clear sky events at the terminators in simulations of tidally
locked Earth-like planets. A feedback between dayside cloud radiative effects,
incoming stellar radiation and heating, and the dynamical state of the
atmosphere, especially the zonal wavenumber-1 Rossby wave identified in past
work on tidally locked planets, leads to oscillations in Rossby wave phase
speeds and in the position of Rossby gyres and results in advection of clouds
to or away from the planet's eastern terminator. We study this oscillation in
simulations of Proxima Centauri b, TRAPPIST 1-e, and rapidly rotating versions
of these worlds located at the extreme inner edge of their stars' habitable
zones. We simulate time series of the transit depths of the 1.4 {\mu}m water
feature and 2.7 {\mu}m carbon dioxide feature. The impact of atmospheric
variability on the transmission spectra is sensitive to the structure of the
dayside cloud cover and the location of the Rossby gyres, but none of our
simulations have variability significant enough to be detectable with current
methods.
