Variational and Hamiltonian Formulations of Geophysical Fluids using Split Exterior Calculus

Christopher Eldred 1 Werner Bauer 2
1 AIRSEA - Mathematics and computing applied to oceanic and atmospheric flows
Grenoble INP - Institut polytechnique de Grenoble - Grenoble Institute of Technology, UGA - Université Grenoble Alpes, LJK - Laboratoire Jean Kuntzmann, Inria Grenoble - Rhône-Alpes
2 FLUMINANCE - Fluid Flow Analysis, Description and Control from Image Sequences
IRMAR - Institut de Recherche Mathématique de Rennes, IRSTEA - Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture, Inria Rennes – Bretagne Atlantique
Abstract : Variational and Hamiltonian formulations for geophysical fluids have proven to be a very useful tool in understanding the physics of flows and developing new numerical discretizations, and represent an important aspect of the geometric structure of the equations for geophysical fluid flow. However, the majority of such formulations have been developed using vector calculus: scalars and vectors. Another key aspect of the geometric structure is understanding the equations in terms of differential forms, and in particular straight and twisted differential forms. This arguably began with the work of Enzo Tonti, who developed a classification if of physical quantities into source and configuration variables; which are unambiguously associated with inner-oriented/straight (configuration) and outer-oriented/twisted (source) geometric entities/differential forms. This classification has proven fruitful in electromagnetics and classical mechanics. However, an extension of the idea to compressible fluids was lacking until the development of the split covariant equations by Werner Bauer. This work aims to unify these two aspects of the geometric structure for fluids, by developing variational and Hamiltonian formulations for geophysical fluids using split exterior calculus. A key aspect is that the Hamil-tonian structure gives a natural representation of the topological-metric splitting in the split covariant equations through the Poisson brackets (purely topological equations) and the functional derivatives of the Hamiltonian (metric-dependent equations). These new formulations are illustrated with some specific examples of commonly studied geophysical fluids: the shallow water equations, thermal shallow water equations and the compressible Euler equations.
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Submitted on : Tuesday, October 16, 2018 - 11:22:29 AM
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Christopher Eldred, Werner Bauer. Variational and Hamiltonian Formulations of Geophysical Fluids using Split Exterior Calculus. 2018. ⟨hal-01895935v1⟩

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