Logarithmic correlation functions in two-dimensional turbulence

We consider the correlation functions of two-dimensional turbulence in the presence and absence of a three-dimensional perturbation, by means of conformal field theory. In the presence of three-dimensional perturbation, we show that in the strong coupling limit of a small scale random force, there is some logarithmic factor in the correlation functions of velocity stream functions. We show that the logarithmic conformal field theory c_8,1 describes the 2D-turbulence both in the absence and in the presence of the perturbation. We obtain the energy spectrum E( k) ∼ k^-5.125 ln( k) for perturbed 2D-turbulence and E( k) ∼ k^-5 ln( k) for unperturbed turbulence. Recent numerical simulation and experimental results confirm our prediction.