Velocity statistics for point vortices in the local α-models of turbulence

The velocity fluctuations for point vortex models are studied for the alpha-turbulence equations, which are characterised 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, that is, to consider an infinite set of point vortices on an infinite plane keeping the density of the vortices constant. The consequence of this limit is that the results obtained are independent on the number of point vortices in the system. 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 turbulence, but can be approximated with a Gaussian function in the small velocity limit. The tails of the distribution exhibit a power law behaviour 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.