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Hamiltonian approach to the modeling of internal geophysical waves with vorticity

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Abstract

We examine a simplified model of internal geophysical waves in a rotational 2-dimensional water-wave system, under the influence of Coriolis forces and with gravitationally induced waves. The system consists of a lower medium, bound underneath by an impermeable flat bed, and an upper lid. The two media have a free common interface. Both media have constant density and constant (non-zero) vorticity. By examining the governing equations of the system we calculate the Hamiltonian of the system in terms of its conjugate variables and perform a variable transformation to show that it has canonical Hamiltonian structure. We then linearize the system, determine the equations of motion of the linearized system and calculate the dispersion relation. Finally, limiting cases are examined to recover irrotational and single medium systems as well as an infinite two media system.

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Acknowledgments

The author would like to thank R. Ivanov and E. Prodanov at the Dublin Institute of Technology for their invaluable discussions on matters relating to this article and also thank an anonymous referee for important suggestions which have contributed to improvements in the overall quality of the article.

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  1. School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland

    Alan Compelli

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  1. Alan Compelli
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Correspondence to Alan Compelli.

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Communicated by A. Constantin.

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Compelli, A. Hamiltonian approach to the modeling of internal geophysical waves with vorticity. Monatsh Math 179, 509–521 (2016). https://doi.org/10.1007/s00605-014-0724-1

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  • DOI: https://doi.org/10.1007/s00605-014-0724-1

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