Rayleigh-B\'enard Convection as a Nambu-metriplectic problem

TL;DR

This paper reformulates Rayleigh-Bénard convection within a Nambu-metriplectic framework, extending its geometric understanding by incorporating both conservative and dissipative dynamics through generalized brackets.

Contribution

It introduces a novel Nambu-metriplectic formulation for RBC, combining conservative and dissipative effects in a unified geometric structure.

Findings

Extended Poisson bracket to a Nambu bracket using Casimir functionals

Represented dissipative RBC as a superposition of Nambu and symmetric brackets

Provides a complete geometric picture of RBC dynamics

Abstract

The traditional Hamiltonian structure of the equations governing conservative Rayleigh-B\'enard convection (RBC) is singular, i.e. it's Poisson bracket possesses nontrivial Casimir functionals. We show that a special form of one of these Casimirs can be used to extend the bilinear Poisson bracket to a trilinear generalised Nambu bracket. It is further shown that the equations governing dissipative RBC can be written as the superposition of the conservative Nambu bracket with a dissipative symmetric bracket. This leads to a Nambu-metriplectic system, which completes the geometrical picture of RBC.