  In this paper, we present strong numerical evidences that the incompressible
axisymmetric Euler equations with degenerate viscosity coefficients and smooth
initial data of finite energy develop a potential finite-time locally
self-similar singularity at the origin. An important feature of this potential
singularity is that the solution develops a two-scale traveling wave that
travels towards the origin. The two-scale feature is characterized by the
scaling property that the center of the traveling wave is located at a ring of
radius $O((T-t)^{1/2})$ surrounding the symmetry axis while the thickness of
the ring collapses at a rate $O(T-t)$. The driving mechanism for this potential
singularity is due to an antisymmetric vortex dipole that generates a strong
shearing layer in both the radial and axial velocity fields. Without the
viscous regularization, the $3$D Euler equations develop a sharp front and some
shearing instability in the far field. On the other hand, the Navier-Stokes
equations with a constant viscosity coefficient regularize the two-scale
solution structure and do not develop a finite-time singularity for the same
initial data.
