Continuous representation for shell models of turbulence

TL;DR

This paper develops continuous hydrodynamic models derived from shell models of turbulence, enabling physical space interpretation and analysis of solutions like shocks and blowups, thus bridging discrete shell models and continuous turbulence descriptions.

Contribution

It introduces continuous models for shell turbulence, linking shell solutions to physical space phenomena and analyzing their properties and singularities.

Findings

Dyadic model solutions correspond to shocks in physical space.

Finite-time blowup scenarios are similar to Burgers equation.

Physical space representations of blowup and turbulence dynamics are provided.

Abstract

In this work we construct and analyze continuous hydrodynamic models in one space dimension, which are induced by shell models of turbulence. After Fourier transformation, such continuous models split into an infinite number of uncoupled subsystems, which are all identical to the same shell model. The two shell models, which allow such a construction, are considered: the dyadic (Desnyansky--Novikov) model with the intershell ratio $\lambda = 2^{3/2}$ and the Sabra model of turbulence with $\lambda = \sqrt{2+\sqrt{5}} \approx 2.058$. The continuous models allow understanding various properties of shell model solutions and provide their interpretation in physical space. We show that the asymptotic solutions of the dyadic model with Kolmogorov scaling correspond to the shocks (discontinuities) for the induced continuous solutions in physical space, and the finite-time blowup together with its viscous regularization follow the scenario similar to the Burgers equation. For the Sabra model, we provide the physical space representation for blowup solutions and intermittent turbulent dynamics.