Probability Distribution of a Passive Scalar in Isotropic Turbulence

TL;DR

This paper develops a Hamiltonian framework for understanding the probability distribution of a passive scalar in isotropic turbulence, providing explicit derivations and analyzing the underlying dynamical structure of turbulence models.

Contribution

It introduces a Hamiltonian approach to passive scalar turbulence and applies the modified Prelle-Singer procedure to derive the probability distribution and analyze the dynamical structure.

Findings

Explicit master equation for passive scalar distribution derived

Hamiltonian representation exists only for certain turbulence systems

Dynamical structure of scale equations characterized

Abstract

In this letter, we present developments of the Hamiltonian approach to problems of the probability distribution for a passive scalar in isotropic turbulence, and also considers specific applications of the modified Prelle-Singer procedure to turbulence models. The following key questions are discussed and solved: what is the general dynamical structure of the resulting scale equation permitted by passive scalar turbulence models? What are the general requirements of the relations between canonical variables and the canonical variabes representation for turbulence by using canonical variables? It is shown that the existence of the Haniltonian representation in turbulence is a privilege of only turbulence systems for which the variational principle of least action is impossible The master equation of the probability distribution of a passive scalar in isotropic turbulence can also be deduced explicitly.