Nambu representation of an extended Lorenz model with viscous heating

We consider the Nambu and Hamiltonian representations of Rayleigh–Bénard convection with a nonlinear thermal heating effect proportional to the Eckert number ( E c ). The model that we use is an extension of the classical Lorenz-63 model with four kinematic and six 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 E c . For E c = 0 two independent conserved functions exist; one of these is associated with unavailable potential energy and is also present in the Lorenz-63 truncation. This function C which is a Casimir of the noncanonical Hamiltonian system is used to construct a Nambu representation of the conserved part of the dynamics. The thermal heating effect can be represented either by a second canonical Hamiltonian or as a gradient (metric) system using the time derivative C ̇ 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.