Towards structure preserving atmospheric finite-mode models

A typical problem with the traditional Galerkin approach for the construction of finitemode models is to keep structural properties untouched 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, 12, 243–254 (1960)] 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 symmetries and the generalized Hamiltonian form (Nambu form). In an analog fashion, it is proved that the famous Lorenz–1963 model violates the structural properties of the Saltzman convection equations and hence cannot be considered as maximum simplification of the Rayleigh–Bénard Convection problem. Using a six-component truncation, we show that it is possible to again retain both symmetries and the Nambu representation during discretization. It is demonstrated that the conservative part of this six-component reduction is related to the equations of a Lagrangian top and hence a sound mechanical interpretation of this new model is available.