Motivated by the observed potential vorticity structure of the stratos
pheric polar vortex, we study the dynamics of linear and nonlinear wav
es on a zonal vorticity interface in a two-dimensional barotropic flow
on the surface of a sphere (interfacial Rossby waves). After reviewin
g the linear problem, we determine, with the help of an iterative sche
me, the shapes of steadily propagating nonlinear waves; a stability an
alysis reveals that they are (nonlinearly) stable up to very large amp
litude. We also consider multi-vortex equilibria on a sphere: we exten
d the results of Thompson (1883) and show that a (latitudinal) ring of
point vortices is more unstable on the sphere than in the plane; nota
bly, no more than three point vortices on the equator can be stable. W
e also determine the shapes of finite-area multi-vortex equilibria, an
d reveal additional modes of instability feeding off shape deformation
s which ultimately result in the complex merger of some or all of the
vortices. We discuss two specific applications to geophysical flows: f
or conditions similar to those of the wintertime terrestrial stratosph
ere, we show that perturbations to a polar vortex with azimuthal waven
umber 3 are close to being stationary, and hence are likely to be reso
nant with the tropospheric wave forcing; this is often observed in hig
h-resolution numerical simulations as well as in the ozone data. Secon
dly, we show that the linear dispersion relation for interfacial Rossb
y waves yields a good fit to the phase velocity of the waves observed
on Saturn's 'ribbon.