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A Nonlocal Shallow-Water Model Arising from the Full Water Waves with the Coriolis Effect

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Abstract

In the present study a mathematical model of the equatorial water waves propagating mainly in one direction with the effect of Earth’s rotation is derived by the formal asymptotic procedures in the equatorial zone. Such a model equation is analogous to the Camassa–Holm approximation of the two-dimensional incompressible and irrotational Euler equations and has a formal bi-Hamiltonian structure. Its solution corresponding to physically relevant initial perturbations is more accurate on a much longer time scale. It is shown that the deviation of the free surface can be determined by the horizontal velocity at a certain depth in the second-order approximation. The effects of the Coriolis force caused by the Earth rotation and nonlocal higher nonlinearities on blow-up criteria and wave-breaking phenomena are also investigated. Our refined analysis is approached by applying the method of characteristics and conserved quantities to the Riccati-type differential inequality.

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Acknowledgements

The authors would like to thank the referees for constructive suggestions and comments. The work of Gui is supported in part by the NSF-China under the Grant Nos. 11571279, 11331005, and the Foundation FANEDD-201315. The work of Liu is supported in part by the Simons Foundation Grant-499875.

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Authors and Affiliations

  1. School of Mathematics, Northwest University, Xi’an, People’s Republic of China

    Guilong Gui

  2. Department of Mathematics, University of Texas at Arlington, Arlington, USA

    Yue Liu & Junwei Sun

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  1. Guilong Gui
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  2. Yue Liu
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  3. Junwei Sun
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Correspondence to Yue Liu.

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Communicated by A. Constantin

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Gui, G., Liu, Y. & Sun, J. A Nonlocal Shallow-Water Model Arising from the Full Water Waves with the Coriolis Effect. J. Math. Fluid Mech. 21, 27 (2019). https://doi.org/10.1007/s00021-019-0432-7

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