  In this paper, Beltrami vector fields in several orthogonal coordinate
systems are obtained analytically and numerically. Specifically, axisymmetric
incompressible inviscid steady state Beltrami (Trkalian) fluid flows are
obtained with the motivation to model flows that have been hypothesized to
occur in tornadic flows. The studied coordinate systems include those that
appear amenable to modeling such flows: the cylindrical, spherical,
paraboloidal, and prolate and oblate spheroidal systems. The usual Euler
equations are reformulated using the Bragg--Hawthorne equation for the stream
function of the flow, which is solved analytically or numerically in each
coordinate system under the assumption of separability of variables. Many of
the obtained flows are visualized via contour plots of their stream functions
in the $rz$-plane. Finally, the results are combined to provide a qualitative
quasi-static model for a progression of flows that one might see in the process
of a vortex breakdown. The results in this paper are equally applicable in
electromagnetics, where the equivalent concept is that of a force-free magnetic
field.
