A new hierarchically-structured n-dimensional covariant form of rotating equations of geophysical fluid dynamics

  • Werner BauerEmail author
Original Paper


We introduce n-dimensional equations of geophysical fluid dynamics (GFD) valid on rotating n-dimensional manifolds. Moreover, by using straight and twisted differential forms and an auxiliary velocity field, we introduce hierarchically-structured equations of (geophysical) fluid dynamics in which the equations are split into metric-free and metric-dependent parts. For these sets of equations we provide representations in local coordinate charts and we show that they conserve potential vorticity and that Kelvin’s circulation theorem holds. As such general n-dimensional formulations do not exist in vector calculus, we provide for both covariant and vector-invariant equations a representation on a rotating coordinate frame in an Euclidean space and compare these representations. This study reveals, among others, that the prognostic variables, described by straight and twisted differential forms, are independent of both metric and orientation. This makes them perfect descriptors of the fluid’s quantities of interest, as they assign, analogously to physical measurement devices, real valued numbers to finite distances, areas, or volumes. This is not the case for prognostic variables described by (vector) proxy fields, as they depend on metric and orientation. The new structuring reveals also important geometrical features of the equations of GFD. For instance, the dimensioned differential forms display the geometric nature of the fluid’s characteristics, while the equations’ structuring illustrates how the metric-free momentum and continuity equations geometrically interact and how they are connected by the metric-dependent Hodge star and Riemannian lift. Besides this geometric insight, this structuring has also practical benefits: since equations containing only topological structure are less complicated to implement and can be integrated exactly, our results may contribute to more efficient and accurate discretizations.


Rotating Euler’s equations Covariant equations Covariant split Euler’s equations of GFD Geophysical fluid dynamics Differential geometry  Exterior calculus 

Mathematics Subject Classification

Primary: 76U05 76A02 58J45 Secondary: 76M10 53Z99 



The author thanks Jörn Behrens for his support and his valuable comments, as well as the two reviewers and F. Guglielmo for their valuable input. The author wishes to acknowledge funding by the Center for Earth System Research and Sustainability CEN at the University of Hamburg, Germany.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.CliSAP/CEN (Center for Earth System Research and Sustainability)University of HamburgHamburgGermany

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