5 research outputs found
Nambu representation of an extended Lorenz model with viscous heating
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.Comment: 15 pages, no figur
Symmetry Analysis of Barotropic Potential Vorticity Equation
Recently F. Huang [Commun. Theor. Phys. V.42 (2004) 903] and X. Tang and P.K.
Shukla [Commun. Theor. Phys. V.49 (2008) 229] investigated symmetry properties
of the barotropic potential vorticity equation without forcing and dissipation
on the beta-plane. This equation is governed by two dimensionless parameters,
and , representing the ratio of the characteristic length scale to
the Rossby radius of deformation and the variation of earth' angular rotation,
respectively. In the present paper it is shown that in the case there
exists a well-defined point transformation to set . The
classification of one- and two-dimensional Lie subalgebras of the Lie symmetry
algebra of the potential vorticity equation is given for the parameter
combination and . Based upon this classification, distinct
classes of group-invariant solutions is obtained and extended to the case
.Comment: 6 pages, release version, added reference for section