  Tidally locked exoplanets likely host global atmospheric circulations with a
superrotating equatorial jet, planetary-scale stationary waves and
thermally-driven overturning circulation. In this work, we show that each of
these features can be separated from the total circulation by using a Helmholtz
decomposition, which splits the circulation into rotational (divergence free)
and divergent (vorticity free) components. This technique is applied to the
simulated circulation of a terrestrial planet and a gaseous hot Jupiter. For
both planets, the rotational component comprises the equatorial jet and
stationary waves, and the divergent component contains the overturning
circulation. Separating out each component allows us to evaluate their spatial
structure and relative contribution to the total flow. In contrast with
previous work, we show that divergent velocities are not negligible when
compared with rotational velocities, and that divergent, overturning
circulation takes the form of a single, roughly isotropic cell that ascends on
the day-side and descends on the night-side. These conclusions are drawn for
both the terrestrial case and the hot Jupiter. To illustrate the utility of the
Helmholtz decomposition for studying atmospheric processes, we compute the
contribution of each of the circulation components to heat transport from day-
to night-side. Surprisingly, we find that the divergent circulation dominates
day-night heat transport in the terrestrial case and accounts for around half
of the heat transport for the hot Jupiter. The relative contributions of the
rotational and divergent components to day-night heat transport are likely
sensitive to multiple planetary parameters and atmospheric processes, and merit
further study.
