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Variational and Hamiltonian Formulations of Geophysical Fluids using Split Exterior Calculus

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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 in the language of vector calculus: scalars and vectors. Another key aspect of the geometric structure is a representation using split exterior calculus: straight and twisted differential forms. This arguably began with the work of Enzo Tonti, who developed a classification of physical quantities into source and configuration variables; which are unambiguously associated with inner-oriented (configuration) and outer-oriented (source) geometric entities, which are themselves associated with straight (inner) and twisted (outer) differential forms. Such a classification has proven fruitful in various areas of classical mechanics, such as electrodynamics, solid mechanics and and some aspects of fluid dynamics. However, an extension of the idea to compressible fluids was lacking until the development of the split covariant equations by Werner Bauer. The current 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 Hamiltonian 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). Additionally, the Lagrangian and Hamiltonian are seen to consist of terms that are pairings between straight and twisted forms. 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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Dates and versions

hal-01895935 , version 1 (16-10-2018)
hal-01895935 , version 2 (04-12-2018)

Identifiers

  • HAL Id : hal-01895935 , version 2

Cite

Christopher Eldred, Werner Bauer. Variational and Hamiltonian Formulations of Geophysical Fluids using Split Exterior Calculus. 2018. ⟨hal-01895935v2⟩
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