  It is known that a localized vortex can have two specific mechanisms of
interaction with the ambient flow. The first mechanism is associated with
acoustic radiation, which is accompanied by a loss of energy and causes
instability in the case of negative energy of vortex disturbances. The second
is a Miles mechanism of interaction of the vortex core oscillations with
disturbances in the vicinity of the critical layer (where the phase velocity of
the disturbances coincides with the velocity of the mean flow), accompanied by
an energy flux from the critical layer vicinity, which leads to damping in the
case of negative energy of the oscillations. For the first time, the flow with
both of these mechanisms is considered. The problem is solved from the first
principles. It is shown that the Miles mechanism can completely suppress
acoustic instability, however, in the case of a stronger loss of energy due to
acoustic radiation, acoustic instability will dominate. The role of various
parameters is analyzed and a quantitative criterion for the acoustic
instability of a vortex with a smoothed vorticity profile is obtained.
