Passive scalar spectrum in the viscous-convective range in two-dimensional steady turbulence

The spectrum of a passive scalar field in the viscous-convective range convected by two-dimensional steady turbulence was studied using the Lagrangian Renormalized Approximation (LRA) and high resolution DNS with a resolution up to 40962. It was found that the scalar spectrum F(k) using the LRA scales as k-1 in the range, and decays Gaussianly in the far diffusive range, and these wavenumber dependencies in both ranges agree with the DNS data. A new Lagrangian closure approximation that is invariant under random Galilean transformation and random uniform rotation is proposed. The approximation makes better quantitative predictions than the LRA and yields F(k) = CVC(ν/)1/2k-1 with CVC = 5.25 in the viscous-convective range, which is in good agreement with the DNS value CVC = 5.11. The importance of molecular diffusion in the far diffusive range when the viscous-convective range exists is also discussed