Minimal atmospheric finite-mode models preserving symmetry and generalized Hamiltonian structures

Details Minimal atmospheric finite-mode models preserving symmetry and generalized Hamiltonian structures Minimal atmospheric finite-mode models preserving symmetry and generalized Hamiltonian structures

2009-09-10

arxiv.org/abs/0909.1957v3

Physics

Atmospheric and Oceanic Physics

18 pages; extended and corrected version; added references

Scientific paper

A typical problem with the conventional Galerkin approach for the construction of finite-mode models is to keep structural properties unaffected in the process of discretization. We present two examples of finite-mode approximations that in some respect preserve the geometric attributes inherited from their continuous models: a three-component model of the barotropic vorticity equation known as Lorenz' maximum simplification equations [Tellus, \textbf{12}, 243--254 (1960)] and a six-component model of the two-dimensional Rayleigh--B\'{e}nard convection problem. It is reviewed that the Lorenz--1960 model respects both the maximal set of admitted point symmetries and an extension of the noncanonical Hamiltonian form (Nambu form). In a similar fashion, it is proved that the famous Lorenz--1963 model violates the structural properties of the Saltzman equations and hence cannot be considered as the maximum simplification of the Rayleigh--B\'{e}nard convection problem. Using a six-component truncation, we show that it is again possible retaining both symmetries and the Nambu representation in the course of discretization. The conservative part of this six-component reduction is related to the Lagrange top equations. Dissipation is incorporated using a metric tensor.

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