  General circulation models of the atmosphere of hot Jupiters have shown the
existence of a supersonic eastward equatorial jet. In this paper, we
investigate the effects of compressibility on the atmospheric dynamics by
solving the standard Euler equations. This is done by means of a series of
simulations performed in the framework of the equatorial beta-plane
approximation using the finite volume shock-capturing code RAMSES. At low
resolution, we recover the classical results described in the literature: we
find a strong and steady supersonic equatorial jet of a few km/s that displays
no signature of shocks. We next show that the jet zonal velocity depends
significantly on the grid meridional resolution. When that resolution is fine
enough to properly resolve the jet, the latter is subject to a Kelvin-Helmholtz
instability. The jet zonal mean velocity displays regular oscillations with a
typical timescale of few days and a significant amplitude of about 15% of the
jet velocity. We also find compelling evidence for the development of a
vertical shear instability at pressure levels of a few bars. It seems to be
responsible for an increased downward kinetic energy flux, significantly
affecting the temperature of the deep atmosphere, and appears to act as a form
of drag on the equatorial jet. This instability also creates velocity
fluctuations that propagate upward and steepen into weak shocks at pressure
levels of a few mbars. We conclude that hot Jupiter equatorial jets are
potentially unstable to both a barotropic Kelvin-Helmholtz instability and a
vertical shear instability. Upon confirmation using more realistic models, both
instabilities could result in significant time variability of the atmospheric
winds, may provide a small scale dissipation mechanism in the flow, and might
have consequences for the internal evolution of hot Jupiters.
