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Nambu representation of an extended Lorenz model with viscous heating

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Blender, R., & Lucarini, V. (2013). Nambu representation of an extended Lorenz model with viscous heating. PHYSICA D-NONLINEAR PHENOMENA, 243(1), 86-91. doi:10.1016/j.physd.2012.09.007.

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 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 Ec. For Ec = 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. (c) 2012 Elsevier B.V. All rights reserved.