  In this paper, we present strong numerical evidences that the $3$D
axisymmetric Navier-Stokes equations with degenerate variable diffusion
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 property that the center of the traveling wave approaches
to the origin at a slower rate than the rate of the collapse of the
singularity. The driving mechanism for this potential singularity is due to two
antisymmetric vortex dipoles that generate a strong shearing layer in both the
radial and axial velocity fields, which transport the solution first towards
$z=0$ and then towards the symmetry axis $r=0$. The initial condition is
designed in such a way that it generates a positive feedback loop that enforces
a strong nonlinear alignment of vortex stretching, leading to a stable locally
self-similar blowup at the origin. We perform careful resolution study and
asymptotic scaling analysis to provide further support of the potential finite
time locally self-similar blowup.
