  The target space of the non-linear $\sigma$-model is a Riemannian manifold.
Although it can be any Riemannian metric, there are certain physically
interesting geometries which are worth to study. Here, we numerically evolve
the time-symmetric foliations of a family of spherically symmetric
asymptotically flat wormholes under the $1$-loop renormalization group flow of
the non-linear $\sigma$-model, the Ricci flow, and under the $2$-loop
approximation, RG-2 flow. We rely over some theorems adapted from the compact
case for studying the evolution of different wormhole types, specially those
with high curvature zones. Some metrics expand and others contract at the
beginning of the flow, however, all metrics pinch-off at a certain time. This
is related with the fact that the flow does not converge to a fixed point when
its starting geometry is the spatial sections of a Morris-Thorne wormhole, and
therefore the corresponding non-linear $\sigma$-model is
non-integrable/renormalizable. We present a numerical study of the evolution of
wormhole singularities in three dimensions extending the theoretical
estimations. Finally, we compute the evolution of the Hamilton's entropy and
the Brown-York energy.
