  We present three-dimensional atmospheric circulation models of a hypothetical
"warm Jupiter" planet, for a range of possible obliquities from 0-90 degrees.
We model a Jupiter-mass planet on a 10-day orbit around a Sun-like star, since
this hypothetical planet sits at the boundary between planets for which we
expect that tidal forces should have aligned their rotation axes with their
orbital axes (i.e., ones with zero obliquity) and planets whose timescale for
tidal alignment is longer than the typical age of an exoplanet system. In line
with observational progress, which is pushing atmospheric characterization to
planets on longer orbital periods, we calculate the observable signatures of
obliquity for a transiting warm Jupiter: in orbital phase curves of thermal
emission and in the hemispheric flux gradients that could be measured by
eclipse mapping. For both of these predicted measurements, the signal that we
would see depends strongly on our viewing geometry relative to the orientation
of the planet's rotation axis, and we thoroughly identify the degeneracies that
result. We compare these signals to the predicted sensitivities of current and
future instruments and determine that the James Webb Space Telescope should be
able to constrain the obliquities of nearby warm Jupiters to be small (if less
than or equal to 10 degrees) or to directly measure them if significantly
non-zero (greater than or equal to 30 degrees), using the technique of eclipse
mapping. For a bright target and assuming photon-limited precision, this could
be done with a single secondary eclipse observation.
