Nambu representation of an extended Lorenz model with viscous heating

TL;DR

This paper develops a Nambu and Hamiltonian framework for an extended Lorenz-63 model incorporating viscous heating, revealing how thermal effects influence energy conservation and dynamics in Rayleigh-Benard convection.

Contribution

It introduces a Nambu representation for a Lorenz extension with viscous heating, highlighting the role of Casimir functions and Hamiltonian structures in the model.

Findings

Viscous heating impacts the energy budget and Lorenz energy cycle.

Conserved Hamiltonian exists for arbitrary Eckert number.

Casimir functions facilitate Nambu representation of the system.

Abstract

We consider the Nambu and Hamiltonian representations of Rayleigh-Benard convection with a nonlinear thermal heating effect proportional to the Eckert number (Ec). The model we use is an extension of the classical Lorenz-63 model with 4 kinematic and 6 thermal degrees of freedom. The conservative parts of the dynamical equations which include all nonlinearities satisfy Liouville's theorem and permit a conserved Hamiltonian H for arbitrary Ec. For Ec=0 two independent conserved Casimir functions exist, one of these is associated with unavailable potential energy and is also present in the Lorenz-63 truncation. This Casimir C is used to construct a Nambu representation of the conserved part of the dynamical system. The thermal heating effect can be represented either by a second canonical Hamiltonian or as a gradient (metric) system using the time derivative of the Casimir. The results demonstrate the impact of viscous heating in the total energy budget and in the Lorenz energy cycle for kinetic and available potential energy.