  Geophysical research has focused on flows, such as ocean currents, as two
dimensional. Two dimensional point or blob vortex models have the advantage of
having a Hamiltonian, whereas 3D vortex filament or tube systems do not
necessarily have one, although they do have action functionals. On the other
hand, certain classes of 3D vortex models called nearly parallel vortex
filament models do have a Hamiltonian and are more accurate descriptions of
geophysical and atmospheric flows than purely 2D models, especially at smaller
scales. In these ``quasi-2D'' models we replace 2D point vortices with vortex
filaments that are very straight and nearly parallel but have Brownian
variations along their lengths due to local self-induction. When very straight,
quasi-2D filaments are expected to have virtually the same planar density
distributions as 2D models. An open problem is when quasi-2D model statistics
behave differently than those of the related 2D system and how this difference
is manifested. In this paper we study the nearly parallel vortex filament model
of Klein, Majda, Damodaran in statistical equilibrium. We are able to obtain a
free-energy functional for the system in a non-extensive thermodynamic limit
that is a function of the mean square vortex position $R^2$ and solve
\emph{explicitly} for $R^2$. Such an explicit formula has never been obtained
for a non-2D model. We compare the results of our formula to a 2-D formula of
\cite{Lim:2005} and show qualitatively different behavior even when we disallow
vortex braiding. We further confirm our results using Path Integral Monte Carlo
(Ceperley (1995)) \emph{without} permutations and that the Klein, Majda,
Damodaran model's asymptotic assumptions \emph{are valid} for parameters where
these deviations occur.
