  We consider the incompressible axisymmetric Navier-Stokes equations with
swirl as an idealized model for tornado-like flows. Assuming an infinite vortex
line which interacts with a boundary surface resembles the tornado core, we
look for stationary self-similar solutions of the axisymmetric Euler and
axisymmetric Navier-Stokes equations. We are particularly interested in the
connection of the two problems in the zero-viscosity limit. First, we construct
a class of explicit stationary self-similar solutions for the axisymmetric
Euler equations. Second, we consider the possibility of discontinuous solutions
and prove that there do not exist self-similar stationary Euler solutions with
slip discontinuity. This nonexistence result is extended to a class of flows
where there is mass input or mass loss through the vortex core. Third, we
consider solutions of the Euler equations as zero-viscosity limits of solutions
to Navier-Stokes. Using techniques from the theory of Riemann problems for
conservation laws, we prove that, under certain assumptions, stationary
self-similar solutions of the axisymmetric Navier-Stokes equations converge to
stationary self-similar solutions of the axisymmetric Euler equations as
$\nu\to0$. This allows to characterize the type of Euler solutions that arise
via viscosity limits.
