  The velocity fluctuations for point vortex models are studied for the
{\alpha}-turbulence equations, which are characterized by a fractional
Laplacian relation between active scalar and the streamfunction. In particular,
we focus on the local dynamics regime. The local dynamics differ from the
well-studied case of 2D turbulence as it allows to consider the true
thermodynamic limit. This limit is not defined for 2D turbulence. We show an
analytical form of the probability density distribution of the velocity
fluctuations for different degrees of locality. The central region of the
distribution is not Gaussian, in contrast to the case of 2D turbulence, but can
be approximated with a Gaussian function in the small velocity limit. The tails
of the distribution exhibit a power law behavior and self similarity in terms
of the density variable. Due to the thermodynamic limit, both the Gaussian
approximation for the core and the steepness of the tails are independent on
the number of point vortices, but depend on the {\alpha}-model. We also show
the connection between the velocity statistics for point vortices uniformly
distributed, in the context of the {\alpha}-model in classical turbulence, with
the velocity statistics for point vortices non-uniformly distributed. Since the
exponent of the power law depends just on {\alpha}, we test the power law
approximation obtained with the point vortex approximation, by simulating full
turbulent fields for different values of {\alpha} and we compute the
correspondent probability density distribution for the absolute value of the
velocity field. These results suggest that the local nature of the turbulent
fluctuations in the ocean or in the atmosphere might be deduced from the shape
of the tails of the probability density functions.
