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

TL;DR

This paper develops a vortex-based approximation to the instanton equations in 2D turbulence, deriving evolution equations for elliptical vortices and linking vortex merging to turbulence cascades.

Contribution

It introduces a Gaussian vortex approximation to the instanton formalism, providing a new variational approach to analyze vortex dynamics in 2D turbulence.

Findings

Vortex merging determines the extremal action for two-point statistics.

Derived evolution equations for elliptical vortices.

Linked vortex dynamics to the inverse cascade process.

Abstract

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.