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

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 E.N. Lorenz, Maximum simplification of the dynamic equations, Tellus 12 (3) (1960) 243-254] and a six-component model of the two-dimensional Rayleigh-Bénard 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-Bénard convection problem. Using a six-component truncation, we show that it is again possible to retain 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.