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Gaussian Vortex Approximation to the Instanton Equations of two-dimensional Turbulence

We investigate two-dimensional turbulence within the Instanton formalism which determines the most probable field in a stochastic classical field theory starting from the Martin-Siggia-Rose path integral. We perform an approximate analysis of these equations based on a variational ansatz using elliptical vortices. The result are evolution equations for the positions and the shapes of the vortices. We solve these ordinary differential equations numerically. The extremal action for the two-point statistics is determined by the merging of two elliptical vortices. We discuss the relationship of this dynamical system to the inverse cascade process of two-dimensional turbulence.