Reassessment of the classical closures for scalar turbulence

In deducing the consequences of the Direct Interaction Approximation, Kraichnan was sometimes led to consider the properties of special classes of nonlinear interactions in degenerate triads in which one wavevector is very small. Such interactions can be described by simplified models closely related to elementary closures for homogeneous isotropic turbulence such as the Heisenberg and Leith models. These connections can be exploited to derive considerably improved versions of the Heisenberg and Leith models that are only slightly more complicated analytically. This paper applies this approach to derive some new simplified closure models for passive scalar advection and investigates the consistency of these models with fundamental properties of scalar turbulence. Whereas some properties, such as the existence of the Kolmogorov–Obukhov range and the existence of thermal equilibrium ensembles, follow the velocity case closely, phenomena special to the scalar case arise when the diffusive and viscous effects become important at different scales of motion. These include the Batchelor and Batchelor–Howells–Townsend ranges pertaining, respectively, to high and low molecular Schmidt number. We also consider the spectrum in the diffusive range that follows the Batchelor range. We conclude that improved elementary models can be made consistent with many nontrivial properties of scalar turbulence, but that such models have unavoidable limitations.