Universal prandtl number in two-dimensional Kraichnan-Batchelor turbulence

We evaluate the universal turbulent Prandtl numbers in the energy and enstrophy regimes of the Kraichnan-Batchelor spectra of two-dimensional turbulence using a self-consistent mode-coupling formulation coming from a renormalized perturbation expansion coupled with dynamic scaling ideas. The turbulent Prandtl number is found to be exactly unity in the (logarithmic) enstrophy regime, where the theory is infrared marginal. In the energy regime, the theory being finite, we extract singularities coming from both ultraviolet and infrared ends by means of Laurent expansions about these poles. This yields the turbulent Prandtl number sigma approximate to 0.9 in the energy regime.