The diffusion model for turbulent energy transfer proposed by Leith is reconsidered from the viewpoint of Markovianized analytical closures based on the direct interaction approximation. We show that the Leith diffusion model represents a subset of the nonlinear interactions; making this connection to analytical closure suggests significant improvements of the Leith model without significantly increasing its analytical complexity. Similar ideas also lead to improved versions of the classical Kovasznay and Heisenberg models. The new models are applied to dissipation range dynamics, the “bottleneck” phenomenon, and the existence of thermalized tails in the truncated Euler equations. As an example of transient spectral dynamics, the models are applied to the development of a Kolmogorov spectrum under steady forcing.