Reassessment of the classical turbulence closures: the Leith diffusion model

Abstract

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.

Keywords:

• isotropic turbulence
• turbulence closures
• Leith diffusion

Acknowledgments

We would like to acknowledge the interesting and useful comments of the referees which helped clarify some important issues.

Notes

¹Julian Scott (private communication) has observed that this failure may only be apparent for the Kovasznay model: since the inviscid Kovasznay model is a first order nonlinear partial differential equation for E(k,t) in conservation form, it can be formulated to admit shock discontinuities. These discontinuous solutions can certainly propagate into unexcited regions of wavenumber space; but needless to say, this means of populating unexcited wavenumbers is very far from plausible.