The equality of the radial extension with the Rossby radius for the tropical cyclone

The two dimensional approximation to the atmospheric intense, large scale, vortex (the tropical cyclone at stationarity) allows the examination of basic properties and in particular the distributon of the vorticity. At stationarity the amount of energy flowing from thermodynamic processes to cover viscous coupling to environment and friction loss represents a small fraction of the mechanical energy stored in the fluid vortical flow. Then the mechanical aspect is dominant and the vorticity will evolve to one of a set of universal spatial distributions. In order to identify the universal asymptotic states of the 2D atmospheric vortex one should develop a treatment similar to that for the 2D ideal (Euler) fluid. The Euler fluid dynamics is equivalently described by a system of discrete point-like vortices interacting in plane by a long range (logarithmic) potential generated by themselves. This discrete model can be formalized in terms of a field theory and the asymptotic stationary states are rigorously shown to verify the sinh-Poisson equation. The major result is deeper: the stationary states exist due to a property called "self-duality", which characterizes all known coherent structures in the nature. The same procedure has been applied to atmosphere: with the difference that now we have an intrinsic length, the Rossby radius. There is an equivalent discrete system of point-like vortices but the interaction potential is short-range. The field theoretical formalism shows that the self-duality is not possible, but cuasi-stationary structures exist, as solutions of the equation (Delta) psi+(1/2)sinh(psi)[cosh(psi) – 1]=0 where psi is the streamfunction. In the present work we discuss a particular consequence of the field theoretical formalism: the radial extension of the atmospheric vortex at stationarity is approximately the Rossby radius. This ensures the maximum of stability (which however is not absolute) of the atmospheric vortex. The origin of this equality R_o (= Rossby radius) = R_{max} (= radial extension of the vortex) is a specific result of the field theoretical (FT) formalism: the mass of the vector boson is equal to the mass of the scalar particle. This has an interpretation in terms of FT: the vector boson gets a mass via the Higgs mechanism and this is due to the presence of a condensate of scalar field at spatial infinity. This condensate is actually the vorticity of the earth rotation. Formally the mass obtained by the vector boson is 1/R_{o} = f^2/(kappa) where f is the Coriolis frequency (assumed fixed) and (kappa) is the coefficient of the Chern-Simons term in the FT Lagrangian. This is the spatial range of interaction between point-like elements of vorticity (which in FT is mediated by vector bosons). On the other hand the scalar particle is actually the cuasi-stable "matter" excitation, i.e. the vortex, i.e. the tropical cyclone vortex. The spatial decay of the vortical solution is the "mass" (1/R_(max)) of the scalar particle excitation. The FT shows that the two masses are equal. Certainly this equality R_o = R_{max} is an approximation of a real situation. We will comment on the meaning of finding departures from this equality. Essentially this means that the cuasi-stationary states have not yet been reached.