The stability of continuously stratified vortices
with large displacement of isopycnal
surfaces on the f-plane is examined both
analytically and numerically. Using an
appropriate asymptotic set of equations, we demonstrated that
sufficiently large
vortices (i.e. those with small values of the Rossby number)
are unstable. Remarkably,
the growth rate of the unstable disturbance is a function of
the spatial coordinates. At
the same time, the corresponding boundary-value problem for
normal modes has no
smooth square-integrable solutions, which would normally
be regarded as stability.
We conclude that (potentially) stable vortices can be
found only among ageostrophic
vortices. Since this assumption cannot be verified
analytically due to complexity of the
primitive equations, we verify it numerically for the
particular case of two-layer stratification.