  Using a 3D general circulation model (GCM), we investigate the sensitivity of
the climate of tidally-locked Earth-like exoplanets, Trappist-1e and Proxima
Centauri b, to the choice of a convection parameterization. Compared to a
mass-flux convection parameterization, a simplified convection adjustment
parameterization leads to a $>$60% decrease of the cloud albedo, increasing the
mean day-side temperature by $\approx$10 K. The representation of convection
also affects the atmospheric conditions of the night side, via a change in
planetary-scale wave patterns. As a result, using the convection adjustment
scheme makes the night-side cold traps warmer by 17-36 K for the planets in our
simulations. The day-night thermal contrast is sensitive to the representation
of convection in 3D GCM simulations, so caution should be taken when
interpreting emission phase curves. The choice of convection treatment,
however, does not alter the simulated climate enough to result in a departure
from habitable conditions, at least for the atmospheric composition and
planetary parameters used in our study. The near-surface conditions both in the
Trappist-1e and Proxima b cases remain temperate, allowing for an active water
cycle. We further advance our analysis using high-resolution model experiments,
in which atmospheric convection is simulated explicitly. Our results suggest
that in a hypothetical global convection-permitting simulation the surface
temperature contrast would be higher than in the coarse-resolution simulations
with parameterized convection. In other words, models with parameterized
convection may overestimate the inter-hemispheric heat redistribution
efficiency.
