  Tidally locked gas giants are typically in several-day orbits, implying a
modest role for rotation in the atmospheric circulation. Nevertheless, there
exist a class of gas-giant, highly irradiated objects---brown dwarfs orbiting
white dwarfs in extremely tight orbits---whose orbital and hence rotation
periods are as short as 1-2 hours. Phase curves and other observations have
already been obtained for this class of objects, raising fundamental questions
about the role of increasing planetary rotation rate in controlling the
circulation. So far, most modeling studies have investigated rotation periods
exceeding a day, as appropriate for typical hot Jupiters. Here we investigate
atmospheric circulation of tidally locked atmospheres with decreasing rotation
periods down to 2.5 hours. With decreasing rotation period, the width of the
equatorial eastward jet decreases, consistent with the narrowing of the
equatorial waveguide due to a decrease of the equatorial deformation radius.
The eastward-shifted equatorial hot spot offset decreases accordingly, and the
off-equatorial westward-shifted hot areas become increasingly distinctive. At
high latitudes, winds become weaker and more rotationally dominated. The
day-night temperature contrast becomes larger due to the stronger influence of
rotation. Our simulated atmospheres exhibit variability, presumably caused by
instabilities and wave interactions. Unlike typical hot Jupiter models, thermal
phase curves of rapidly rotating models show a near alignment of peak flux to
secondary eclipse. This result helps to explain why, unlike hot Jupiters, many
brown dwarfs orbiting white dwarfs exhibit IR flux peaks aligned with secondary
eclipse. Our results have important implications for understanding
fast-rotating, tidally locked atmospheres.
