WAVE AND VORTEX DYNAMICS ON THE SURFACE OF A SPHERE

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.