Scaling in an Ensemble of Stochastic Forced Point Vortices

Abstract

Onsager's point vortex model of two dimensional turbulence is extended by the inclusion of time dependent vortex circulations. If the time dependence of the circulations is governed by statistically independent Onstein-Uhlenbeck processes we observe the emergence of scaling regimes for the structure functions of the Eulerian and the Lagrangian velocity increments. Fully developed turbulent flows are flux equilibrium systems leading to selfsimilarity and scaling behaviour of correlation functions. The probability distributions corresponding to the simplest case of idealized, i.e. homogeneous, isotropic, and stationary fully developed flows are unknown although the underlying fluid dynamical equations and its statistical counterparts are well-established [1]. Fluid motions can be treated either from an Eulerian or a Lagrangian point of view. Most analytical theories have been formulated in the Eulerian framework. Point vortex models (see e.g. [2]), which have been extensively investigated especially for the case of two dimensional flows, essentially make use of a Lagrangian description. Since the point vortex equations of an ideal two dimensional fluid, which is not stirred, are of Hamiltonian nature, a statistical treatment based on the microcanonical ensemble can be established. This has been discussed for the first time by Onsager [3]. The purpose of the present Letter is to show that an extension of Onsager's point vortex model, which allows for fluctuating circulations of the point vortices, leads to a state which shows scaling behaviour of the structure functions.