The Coulomb Gas Behaviour of Two Dimensional Turbulence

The long-time large-distance behaviour of free decaying two dimensional turbulence is studied. Stochastic solutions of the Navier-Stokes equation are explicitly shown to follow renormalisation group trajectories. It is proven that solutions of the Navier-Stokes equation asymptotically converge to fixed points which are conformal field theories. A particular fixed point is given by the free Gaussian field with a charge at infinity. The stream function is identified with a vertex operator. It happens that this solution also admits constant $n$-enstrophy fluxes in the asymptotic regime, therefore fulfilling all requirements to represent an asymptotic state of two-dimensional turbulence. The renormalisation basin of attraction of this fixed point consists of a charged Coulomb gas. This Coulomb gas gives an effective description of turbulence.