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Stratified equatorial flows in the \(\beta \)-plane approximation with a free surface

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Abstract

We investigate the exact solutions to the governing equations for the equatorial flows with the associated free surface and rigid bottom boundary conditions in the \(\beta \)-plane approximation which incorporates two considerations of the density stratification. Compared to the spherical coordinates and the cylindrical coordinates, the employment of the \(\beta \)-plane approximation admits that the density can be provided generally. Utilizing the implicit theorem, we present the Bernoulli relation between the pressure imposed on the free surface and the resulting distortion of the surface and we obtain that this relation exhibits the expected monotonicity properties. Finally, we prove that certain flows established by the exact solutions are stable via the short-wavelength stability method and the specific assumption of the density distribution.

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Acknowledgements

The authors are grateful to the referees for their careful reading of the manuscript and their valuable comments.

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Authors and Affiliations

  1. Department of Mathematics, Guizhou University, Guiyang, 550025, Guizhou, China

    Fahe Miao & JinRong Wang

  2. Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynská dolina, Bratislava, 842 48, Slovakia

    Michal Fečkan

  3. Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, Bratislava, 814 73, Slovakia

    Michal Fečkan

Authors
  1. Fahe Miao
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  2. Michal Fečkan
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  3. JinRong Wang
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Correspondence to JinRong Wang.

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Communicated by Adrian Constantin.

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This work is partially supported by the National Natural Science Foundation of China (12161015), Training Object of High Level and Innovative Talents of Guizhou Province ((2016)4006), Major Research Project of Innovative Group in Guizhou Education Department ([2018]012), Guizhou Data Driven Modeling Learning and Optimization Innovation Team ([2020]5016) the Slovak Research and Development Agency under the Contract No. APVV-18-0308, and by the Slovak Grant Agency VEGA Nos. 1/0358/20 and 2/0127/20.

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Miao, F., Fečkan, M. & Wang, J. Stratified equatorial flows in the \(\beta \)-plane approximation with a free surface. Monatsh Math 200, 315–334 (2023). https://doi.org/10.1007/s00605-022-01685-2

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  • DOI: https://doi.org/10.1007/s00605-022-01685-2

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