# Optimal subgrid scheme for shell models of turbulence

## TL;DR

This paper develops a theoretical framework for optimal sub-grid closures in shell models of turbulence, demonstrating that low-order approximations can accurately reproduce large-scale dynamics and properties, with some scale-dependent discrepancies.

## Contribution

It introduces a systematic approach to define and approximate optimal sub-grid closures based on scale correlations, extending ideas from Kolmogorov's hypotheses.

## Key findings

- Low-order closures reproduce known turbulence properties.
- Discrepancies occur near the sub-grid threshold scale.
- Connections to Large Eddy Simulations are discussed.

## Abstract

We discuss a theoretical framework to define an optimal sub-grid closure for shell models of turbulence. The closure is based on the ansatz that consecutive shell multipliers are short-range correlated, following the third hypothesis of Kolmogorov formulated for similar quantities for the original three-dimensional Navier-Stokes turbulence. We also propose a series of systematic approximations to the optimal model by assuming different degrees of correlations across scales among amplitudes and phases of consecutive multipliers. We show numerically that such low-order closures work well, reproducing all known properties of the large-scale dynamics including anomalous scaling. We found small but systematic discrepancies only for a range of scales close to the sub-grid threshold, which do not tend to disappear by increasing the order of the approximation. We speculate that the lack of convergence might be due to a structural instability, at least for the evolution of very fast degrees of freedom at small scales. Connections with similar problems for Large Eddy Simulations of the three-dimensional Navier-Stokes equations are also discussed. Postprint version of the article published on Phys. Rev. E 95, 043108 (2017) DOI: 10.1103/PhysRevE.00.003100