Statistical theory of passive scalar turbulence within the viscous-convective range

An expected statistical law of passive scalar turbulence subjected to high Schmidt numbers is studied with the Hessian-based Lagrangian renormalized approximation (HBLRA), a self-consistent closure theory recently developed for passive scalar turbulence. Without relying on any empirical parameters, HBLRA derives the scalar variance spectrum proportional to k-1 in the viscous-convective range predicted by Batchelor. Thanks to improvements in the physical description, HBLRA yields quantitatively better predictions of the scalar variance spectrum and its universal constant in comparison with pioneering closure theories. Our results suggest that the turbulence motion of only a limited scale range may contribute to the scalar flux within the viscous-convective range. Numerical solutions for finite Schmidt numbers (Sc = 10-100 000) are obtained, where a clear viscous-convective range appears with a well-developed inertial-convective range.