  The linear stability of a rotating, stratified, inviscid horizontal plane
Couette flow in a channel is studied in the limit of strong rotation and
stratification. An energy argument is used to show that unstable perturbations
must have large wavenumbers. This motivates the use of a WKB-approach which, in
the first instance, provides an approximation for the dispersion relation of
the various waves that can propagate in the flow. These are Kelvin waves,
trapped near the channel walls, and inertia-gravity waves with or without
turning points.
  Although, the wave phase speeds are found to be real to all algebraic orders
in the Rossby number, we establish that the flow, whether cyclonic or
anticyclonic, is unconditionally unstable. This is the result of linear
resonances between waves with oppositely signed wave momenta. We derive
asymptotic estimates for the instability growth rates, which are exponentially
small in the Rossby number, and confirm them by numerical computations. Our
results, which extend those of Kushner et al (1998) and Yavneh et al (2001),
highlight the limitations of the so-called balanced models, widely used in
geophysical fluid dynamics, which filter out Kelvin and inertia-gravity waves
and hence predict the stability of the Couette flow. They are also relevant to
the stability of Taylor-Couette flows and of astrophysical accretion discs.
