Nambu calculus in dynamical meteorology

Nambu calculus is a formalism designed for convenient representation of a number of classical mechanical systems. It has originally been developed by Nambu (1973) for the dynamics of discrete mass points; somewhat later, authors applied it to the continuous field equations of dynamical meteorology. These attempts have been promising, specifically in form of the energy-vorticity-theory as developed recently by Névir (1998). The present proposal aims at further developing this new approach through both generalizing its formal foundations and applying it to an extended class of relevant fluid dynamic problems. Goal of this proposal is to show that the new theory, by using the Nambu calculus together with the methods of differential geometry, including Lie group theory, has the potential to unify the principles of dynamic meteorology. One important step will be to examine how the discrete Nambu structures are related to the continuous partial differential equations of fluids. The project will in its theoretical part focus upon the inviscid barotropic vorticity equation, the prototype of nonlinear geophysical fluid dynamics. It possesses, in the framework of energy-vorticity dynamics, a characteristic form, the trilinear Nambu bracket, that reveals the geometric structure of the continuous vorticity equation. An equivalent formalism is applicable to higher-order problems like Rayleigh-Bénard convection which has two unknown functions and includes dissipation; the latter generalization leads to Nambu-metriplectic systems. The project will in its applied part focus upon exploring the potential of the energy-vorticity-theory for running discrete, but structure-preserving, low-order models of fluid dynamics. This task will be done by supplementing the traditional Galerkin methods with the information contained in the Nambu form of the corresponding continuous equations. The financial volume applied for is about EUR 225.000, which includes the salary for 2 PhD students for three years, plus travel expenses and computer resources. Planned duration is 36 months (January 2009 until December 2011). The project will be carried out at the Faculty of Mathematics of the University of Vienna.

Nambu calculus is a formalism for representing the evolution equations of physical systems. Originally developed by Y. Nambu (1973) for discrete systems, it was later extended to encompass the continuous field equations of fluid dynamics. Characteristic property of the Nambu representation is its explicit consideration of the general conservation laws, in particular the conservation of energy and of one vorticity moment (e.g., the square of vorticity, colloquially referred to as enstrophy).A key instrument of this calculus is the Nambu bracket which essentially represents the nonlinear terms in the dynamic equations. The Nambu bracket generalizes the Poisson bracket of classical Hamiltonian mechanics. In the present project we have investigated its role for the vorticity equation. Our main task was to study in detail the theoretical foundation of the Nambu bracket and its practical significance for numerical forecasting. For example, various low-order spectral models were studied; their analysis could be visibly improved with the Nambu representation. Also, conservative grid-point discretizations of the vorticity equation were systematically derived. These structure-preserving numerical schemes are of major importance in weather and climate prediction.The project has demonstrated that using geometric properties of partial differential equations can have a beneficial impact on simulation models of increasing complexity.