  Recent observations near the Galactic Centre have found several molecular
filaments displaying striking helically-wound morphology, which are
collectively known as "molecular tornadoes." We investigate the equilibrium
structure of these molecular tornadoes by formulating a magnetohydrodynamic
model of a rotating, helically magnetized filament. A special analytical
solution is derived where centrifugal forces balance exactly with toroidal
magnetic stress. From the physics of torsional Alfv\'{e}n waves, we derive a
constraint that links the toroidal flux-to-mass ratio and the pitch angle of
the helical field to the rotation laws, which we find to be an important
component in describing molecular tornado structure. The models are compared to
the Ostriker solution for isothermal, non-magnetic, non-rotating filaments. We
find that neither the analytic model nor the Alfv\'{e}n wave model suffer from
unphysical density inversions noted by other authors. A Monte Carlo exploration
of our parameter space is constrained by observational measurements of the
Pigtail Molecular Cloud (Pigtail), Double Helix Nebula (DHN), and Galactic
Centre molecular Tornado (GCT). Observable properties such as the velocity
dispersion, filament radius, linear mass, and surface pressure can be used to
derive three dimensionless constraints for our dimensionless models of these
three objects. A virial analysis of these constrained models is studied for
these three molecular tornadoes. We find that self-gravity is relatively
unimportant, whereas magnetic fields, and external pressure play a dominant
role in the confinement and equilibrium radial structure of these objects.
