  The elliptical instability of a rotating stratified fluid is examined in the
regime of small Rossby number and order-one Burger number corresponding to
rapid rotation and strong stratification. The Floquet problem describing the
linear growth of disturbances to an unbounded, uniform-vorticity elliptical
flow is solved using exponential asymptotics. The results demonstrate that the
flow is unstable for arbitrarily strong rotation and stratification; in
particular, both cyclonic and anticyclonic flows are unstable. The instability
is weak, however, with growth rates that are exponentially small in the Rossby
number. The analytic expression obtained for the growth rate elucidates its
dependence on the Burger number and on the eccentricity of the elliptical flow.
It explains in particular the weakness of the instability of cyclonic flows,
with growth rates that are only a small fraction of those obtained for the
corresponding anticyclonic flows. The asymptotic results are confirmed by
numerical solutions of Floquet problem.
