Motion of a vortex sheet on a sphere with pole vortices
Abstract:
We consider the motion of a vortex sheet on the surface of a unit sphere in the presence of point vortices
fixed on north and south poles.
Analytic and numerical research revealed that a vortex sheet in
two-dimensional space has the following three properties.
First, the vortex sheet is linearly unstable due to
Kelvin-Helmholtz instability. Second, the curvature of the
vortex sheet diverges in finite time. Last, the vortex sheet
evolves into a rolling-up doubly branched spiral, when the equation
of motion is regularized by the vortex method.
The purpose of this article is to
investigate how the curvature of the sphere and the presence of the
pole vortices affect these three properties mathematically and
numerically. We show that some low spectra of disturbance
become linearly stable due to 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
complex structure of many rolling-up spirals.
Conclusion:
Parameters:
Reference:
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