  The large scale atmospheric vortices (tropical cyclones, tornadoes) are
complex physical systems combining thermodynamics and fluid-mechanical
processes. The well known tendency of vorticity to self-organization, an
universal property of the two-dimensional fluids, is part of the full dynamics,
but its description requires particular methods. The general framework for the
thermodynamical and mechanical processes is based on conservation laws while
the vorticity self-organization needs a variational approach. It is difficult
to estimate to what extent the vorticity self-organization (a purely kinematic
process) have influenced the characteristics of the tropical cyclone at
stationarity. If this influence is substantial it is expected that the
stationary state of the tropical cyclone has the same nature as the vortices of
many other systems in nature: ideal (Euler) fluids, superconductors, Bose -
Einstein condensate, cosmic strings, etc.
  In previous works we have formulated a description of the $2D$ vorticity
self-organization in terms of a classical field theory. The field theoretical
(FT) formulation finds that the quasi-coherent form of the atmospheric vortex
(tropical cyclone) at stationarity is an expression of the Self-Duality.
  In the present work we examine a strong property of the tropical cyclone,
which arises in the FT formulation in a natural way: the equality of the masses
of the particles associated to the matter field and respectively to the gauge
field in the FT model is translated into the equality between the maximum
radial extension of the tropical cyclone and the Rossby radius. For the cases
where the FT model is a good approximation we calculate characteristic
quantities of the tropical cyclone and find good comparison with observational
data.
