  Planets in M dwarf stars' habitable zones are likely to be tidally locked
with orbital periods of order tens of days. This means that the effects of
rotation on atmospheric dynamics will be relatively weak, which requires small
horizontal temperature gradients above the boundary layer of terrestrial
atmospheres. An analytically solvable and dynamically consistent model for
planetary climate with only three free parameters can be constructed by making
the weak temperature gradient (WTG) approximation, which assumes temperatures
are horizontally uniform aloft. The extreme numerical efficiency of a WTG model
compared to a 3D general circulation model (GCM) makes it an optimal tool for
Monte Carlo fits to observables over parameter space. Additionally, such
low-order models are critical for developing physical intuition and coupling
atmospheric dynamics to models of other components of planetary climate. The
objective of this paper is to determine whether a WTG model provides an
adequate approximation of the effect of atmospheric dynamics on quantities
likely to be observed over the next decade. To do this we first tune a WTG
model to GCM output for an Earth-like tidally locked planet with a dry, 1 bar
atmosphere, then generate and compare the expected phase curves of both models.
We find that differences between the two models would be extremely difficult to
detect from phase curves using JWST. This result demonstrates the usefulness of
the WTG approximation when used in conjunction with GCMs as part of a modeling
hierarchy to understand the climate of remote planets.
