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Fluids 2017, 2(2), 24; doi:10.3390/fluids2020024

Construction of Hamiltonian and Nambu Forms for the Shallow Water Equations

1
Meteorological Institute, University of Hamburg, Hamburg 20146, Germany
2
Institute of Oceanography, Center for Earth System Research and Sustainability (CEN), University of Hamburg, Hamburg 20146, Germany
*
Author to whom correspondence should be addressed.
Academic Editor: Pavel S. Berloff
Received: 10 March 2017 / Revised: 28 April 2017 / Accepted: 11 May 2017 / Published: 13 May 2017
(This article belongs to the Collection Geophysical Fluid Dynamics)
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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. View Full-Text
Keywords: shallow water; Hamiltonian dynamics; Nambu dynamics shallow water; Hamiltonian dynamics; Nambu dynamics
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Blender, R.; Badin, G. Construction of Hamiltonian and Nambu Forms for the Shallow Water Equations. Fluids 2017, 2, 24.

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