We consider the motion of a vortex sheet on the surface of
a sphere in the presence of point vortices fixed at the
north and south poles. Prior analytical and numerical studies
revealed that a vortex sheet in two-dimensional planar flow
has the following three properties: (1) the vortex sheet is
linearly unstable due to Kelvin-Helmholtz instability,
(2) the curvature of the vortex sheet diverges in finite time,
and (3) the vortex sheet evolves into a rolling-up doubly
branched spiral when the equation of motion is regularized by
the vortex method. The present work investigates how the spherical
curvature and the presence of the pole vortices affects these
three properties. We show that some low wavenumber spectra are
stabilized by the pole vortices, and thus the singularity
formation tends to be delayed. On the other hand however, the
vortex sheet, which is regularized by the vortex method,
acquires a complex structure of many rolling-up spirals.
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