Abstract
A systematic method to derive the Hamiltonian and Nambu form for the shallow water equations using the conservation for energy and potential enstrophy is presented. Different mechanisms, such as vortical flows and emission of gravity waves, emerge from different conservation laws for total energy and potential enstrophy. The equations are constructed using exterior differential forms and self-adjoint operators, and result in the sum of two Nambu brackets—one for the vortical flow and one for the wave-mean flow interaction—and a Poisson bracket representing the interaction between divergence and geostrophic imbalance. The advantage of this approach is that the Hamiltonian and Nambu forms can here be written in a coordinate-independent form.
Highlights
Noncanonical Hamiltonian dynamics is the natural framework for the geometric description of hydrodynamical systems in Eulerian form, and is characterized by the fact that the Poisson operator is singular, with the singularity giving rise to a class of conserved quantities called Casimirs
The approach has been applied to finite-dimensional systems, including the nondissipative Lorenz equations [3] and the dynamics of point-vortices [4,5,6], and it has been extended to infinite dimensional systems by Névir and Blender [7], where the Nambu brackets for ideal hydrodynamics were formulated using enstrophy and helicity as conserved quantities in two and three dimensions, respectively
The main advantage of the method presented here is to simplify the intuitive approach, which is often necessary in the construction of dynamical equations consistent with conservation laws
Summary
Noncanonical Hamiltonian dynamics is the natural framework for the geometric description of hydrodynamical systems in Eulerian form, and is characterized by the fact that the Poisson operator is singular, with the singularity giving rise to a class of conserved quantities called Casimirs. [19] showed that the equations for hydrodynamic systems can be derived in Nambu form purely using conservation laws (CLs) and geometric principles. The dynamics can be described in terms of physical processes if the second integral is replaced by two integrals making use of the squares of vorticity and temperature, which correspond to enstrophy and available potential energy. In the Appendix, we present the Nambu form for a finite-dimensional system obtained by the coefficients of the Fourier expansion of the shallow water equations and describing the interaction between the “slow” vorticity variables and the “fast” gravity waves modes
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