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Lagrangian approach to nonlinear waves in non-dispersive and dispersive rotating shallow water magnetohydrodynamics

Published online by Cambridge University Press:  22 March 2024

Vladimir Zeitlin Open the ORCID record for Vladimir Zeitlin [Opens in a new window]
Vladimir Zeitlin*
Affiliation:
Laboratory of Dynamical Meteorology, Sorbonne University, Ecole Normale Supérieure, CNRS, 24 rue Lhomond, 75005 Paris, France SUSTech International Center for Mathematics, Southern University of Science and Technology, Shenzhen 518055, PR China
*
†Email address for correspondence: zeitlin@lmd.ens.fr
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Abstract

A Lagrangian approach to both hydrostatic non-dispersive in the short-wave range and non-hydrostatic dispersive rotating shallow-water magnetohydrodynamics is developed, and used to analyse weakly and fully nonlinear waves described by the model. Hyperbolic structure in the non-dispersive case is displayed and Riemann invariants are constructed. Characteristic equations are used to establish criteria of breaking and formation of shocks by magneto-gravity waves, and conditions of the appearance of contact discontinuities in Alfvén waves. As in the case of non-magnetic rotating shallow water, rotation cannot prevent breaking. The Lagrangian equations of the model are reduced to a single partial differential ‘master’ equation, which is used to analyse the propagation of weakly nonlinear waves of both families, with or without weak rotation, and with or without weak short-wave dispersion. Corresponding modulation equations are constructed and their main properties sketched. The same master equation is used to obtain fully nonlinear finite-amplitude wave solutions in particular cases of no short-wave dispersion or no rotation.

JFM classification

Geophysical and Geological Flows: Shallow water flows MHD and Electrohydrodynamics: Magnetic fluids Waves/Free-surface Flows: Wave breaking
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 983 , 25 March 2024 , A42
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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