| Abstract: A weak instability mode, associated with phase-locked
counterpropagating coastal Kelvin waves in horizontal anticyclonic shear,
is found in the semigeostrophic (SG) equations for stratified flow in a
channel. This SG instability mode approximates a similar mode found in
the Euler equations in the limit in which particle-trajectory slopes are
much smaller than f/N, where f is the Coriolis frequency
and N > f the buoyancy frequency. Though
weak under normal parameter conditions, this instability mode is of theoretical
interest because its existence accounts for the failure of an Arnol'd-type
stability theorem for the SG equations. In the opposite limit, in which
the particle motion is purely vertical, the Euler equations allow only
buoyancy oscillations with no horizontal coupling. The SG equations, on
the other hand, allow a physically spurious coastal "mirage wave",
so called because its velocity field vanishes despite a nonvanishing disturbance
pressure field. Counterpropagating pairs of these waves can phase-lock
to form a spurious "mirage-wave instability." Closer examination
shows that the mirage wave arises from failure of the SG approximations
to be self-consistent for trajectory slopes greater than approximately
f/N. |