Integrability and Hamiltonian system in isotropic turbulence

TL;DR

This paper explores the Hamiltonian framework for isotropic turbulence decay, revealing a nonlinear ODE linked to the Karman-Howarth equation, demonstrating integrability and nonstandard Hamiltonian structures.

Contribution

It introduces a Hamiltonian approach to isotropic turbulence decay and applies the modified Prelle-Singer procedure to establish integrability and Hamiltonian structures.

Findings

Identifies a nonlinear ODE related to turbulence decay

Shows the existence of time-dependent first integrals

Proves nonstandard Hamiltonian structure and Liouville integrability

Abstract

We present developments of the Hamiltonian approach to problems of the freely decay of isotropic turbulence, and also consider specific applications of the modified Prelle-Singer procedure to isotropic turbulence. It demonstrates that a nonlinear second order ordinary differential equation is intimately related to the self-preserving solution of Karman-Howarth equation, admitting time-dependent first integrals and also proving the nonstandard Hamiltonian structure, as well as the Liouville sense of integrability.