Abstract
We consider a mathematical structure of the mean field equation for nearly parallel vortex filament systems in the periodic broken path model. This system can be regarded as a three-dimensional extension of the point vortex system introduced by Onsager to investigate the 2D turbulence.
In this paper, we show the existence of a solution to the mean field equation of the system by analyzing the variational functional and reveal that this system is equipped with a dual variational structure as in the case of the point vortex system and that two mean field equations in terms of the stream function and the vorticity are derived from a Lagrangian. The mean field equation used in this paper is modified from that in our previous research by considering the boundary condition carefully, thus we get finer results.
