Exact Periodic Solutions of Shells Models of Turbulence

TL;DR

This paper derives exact analytical solutions for the GOY shell model of turbulence using elliptic functions, revealing integrability in small cases and Kolmogorov solutions in the limit, supported by numerical simulations.

Contribution

It provides the first exact solutions for the shell model of turbulence, including recursion relations and stability analysis, advancing understanding of turbulence dynamics.

Findings

Exact solutions for three shells are integrable.

Recursion relations for amplitudes in many shells are derived.

Numerical simulations confirm analytical predictions and reveal chaotic transitions.

Abstract

We derive exact analytical solutions of the GOY shell model of turbulence. In the absence of forcing and viscosity we obtain closed form solutions in terms of Jacobi elliptic functions. With three shells the model is integrable. In the case of many shells, we derive exact recursion relations for the amplitudes of the Jacobi functions relating the different shells and we obtain a Kolmogorov solution in the limit of infinitely many shells. For the special case of six and nine shells, these recursions relations are solved giving specific analytic solutions. Some of these solutions are stable whereas others are unstable. All our predictions are substantiated by numerical simulations of the GOY shell model. From these simulations we also identify cases where the models exhibits transitions to chaotic states lying on strange attractors or ergodic energy surfaces.