Skip to main content
SpringerLink
Log in

On some azimuthal equatorial flows

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

In this paper, we are concerned with the incompressible and inviscid equatorial flows with discontinuous stratification. Firstly, arguing along the lines of Constantin and Johnson (J Phys Oceanogr 46:1935–1945, 2016), Henry and Martin (J Differ Equ 266:6788–6808, 2019, Dyn Partial Differ Equ 15:337–349, 2018, Arch Ration Mech Anal 233:497–512, 2019, Nonlinearity 33:3889–3904, 2020), Martin and Quirchmayr (J Math Phys 60:101505, 2019), Martin (Phys Fluids 33:026602, 2021) and Martin and Quirchmayr (Stud Appl Math 148:1021–1039, 2022) we construct an exact solution to the governing equations of the discontinuous stratified equatorial flows in cylindrical coordinates. Secondly, we prove that the pressure exerted on the free surface defines implicitly the shape of the surface deformation. Finally, we apply continuous pressure along the interface to produce an equation that (implicitly) describes the shape of the interface. Interestingly, it turns out that the interface defining function has infinite regularity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Constantin, A., Johnson, R.S.: An exact, steady, purely azimuthal equatorial flow with a free surface. J. Phys. Oceanogr. 46, 1935–1945 (2016)

    Article  Google Scholar 

  2. Constantin, A., Johnson, R.S.: An exact, steady, purely azimuthal fow as a model for the Antarctic Circumpolar Current. J. Phys. Oceanogr. 46, 3585–3594 (2016)

    Article  Google Scholar 

  3. Hsu, H.-C., Martin, C.I.: Free-surface capillary-gravity azimuthal equatorial flows. Nonlinear Anal. 144, 1–9 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Hsu, H.-C., Martin, C.I.: On the existence of solutions and the pressure function related to the Antarctic Circumpolar Current. Nonlinear Anal. 155, 285–293 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Quirchmayr, R.: A steady, purely azimuthal flow model for the Antarctic Circumpolar Current. Monatsh. Math. 187, 565–572 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chu, J., Ionescu-Kruse, D., Yang, Y.: Exact solution and instability for geophysical waves at arbitrary latitude. Discrete Contin. Dyn. Syst. 39, 4399–4414 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chu, J., Ionescu-Kruse, D., Yang, Y.: Exact solution and instability for geophysical waves with centripetal forces and at arbitrary latitude. J. Math. Fluid Mech. 21, 19 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  8. Miao, F., Fečkan, M., Wang, J.: Stratified equatorial flows in the \(\beta \)-plane approximation with a free surface. Monatsh. Math. (2022). https://doi.org/10.1007/s00605-022-01685-2

    Article  Google Scholar 

  9. Guan, Y., Wang, J., Fečkan, M.: Periodic solutions and Hyers–Ulam stability of atmospheric Ekman flows. Discrete Contin. Dyn. Syst. 41, 1157–1176 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  10. Constantin, A.: An exact solution for equatorially trapped waves. J. Geophys. Res. Oceans 117, C05029 (2012)

    Article  Google Scholar 

  11. Constantin, A.: Some three-dimensional nonlinear equatorial flows. J. Phys. Oceanogr. 43, 165–175 (2013)

    Article  Google Scholar 

  12. Constantin, A., Johnson, R.S.: On the nonlinear, three-dimensional structure of equatorial oceanic flows. J. Phys. Oceanogr. 49, 2029–2042 (2019)

    Article  Google Scholar 

  13. Constantin, A.: On the modelling of equatorial waves. Geophys. Res. Lett. 39, L05602 (2012)

    Article  Google Scholar 

  14. Constantin, A., Johnson, R.S.: The dynamics of waves interacting with the Equatorial Undercurrent. Geophys. Astrophys. Fluid Dyn. 109, 311–358 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Constantin, A., Johnson, R.S.: A nonlinear, three-dimensional model for ocean flows, motivated by some observations of the pacific equatorial undercurrent and thermocline. Phys. Fluids 29, 056604 (2017)

    Article  Google Scholar 

  16. McCreary, J.P.: Modeling equatorial ocean circulation. Ann. Rev. Fluid Mech. 17, 359–409 (1985)

    Article  MATH  Google Scholar 

  17. Constantin, A., Ivanov, R.I.: A Hamiltonian approach to wave-current interactions in two-layer fluids. Phys. Fluids 27, 086603 (2015)

    Article  MATH  Google Scholar 

  18. Constantin, A., Ivanov, R.I., Martin, C.I.: Hamiltonian formulation for wave-current interactions in stratified rotational flows. Arch. Ration. Mech. Anal. 221, 1417–1447 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Constantin, A., Ivanov, R.I.: Equatorial wave-current interactions. Commun. Math. Phys. 370, 1–48 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  20. Martin, C.I.: Some explicit solutions of the three-dimensional Euler equations with a free surface. Math. Ann. (2021). https://doi.org/10.1007/s00208-021-02323-2

    Article  Google Scholar 

  21. Henry, D., Villari, G.: Flow underlying coupled surface and internal waves. J. Differ. Equ. 310, 404–442 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  22. Henry, D., Martin, C.I.: Free-surface, purely azimuthal equatorial flows in spherical coordinates with stratification. J. Differ. Equ. 266, 6788–6808 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  23. Henry, D., Martin, C.I.: Exact, purely azimuthal stratified equatorial flows in cylindrical coordinates. Dyn. Partial Differ. Equ. 15, 337–349 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  24. Henry, D., Martin, C.I.: Exact, free-surface equatorial flows with general stratification in spherical coordinates. Arch. Ration. Mech. Anal. 233, 497–512 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  25. Henry, D., Martin, C.I.: Stratified equatorial flows in cylindrical coordinates. Nonlinearity 33, 3889–3904 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  26. Martin, C.I., Quirchmayr, R.: Explicit and exact solutions concerning the Antarctic Circumpolar Current with variable density in spherical coordinates. J. Math. Phys. 60, 101505 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  27. Fan, L., Shen, S., Chen, Y.: A cylindrical coordinates approach concerning the Antarctic Circumpolar Current. Monatsh. Math. 196, 269–279 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  28. Fedorov, A.V., Brown, J.N.: Equatorial waves. In: Steele, J. (ed.) Encyclopedia of Ocean Sciences, pp. 3679–3695. Academic press, New York (2009)

    Google Scholar 

  29. Martin, C.I.: Azimuthal equatorial flows in spherical coordinates with discontinuous stratification. Phys. Fluids 33, 026602 (2021)

    Article  Google Scholar 

  30. Martin, C.I., Quirchmayr, R.: Exact solutions and internal waves for the Antarctic Circumpolar Current in spherical coordinates. Stud. Appl. Math. 148, 1021–1039 (2022)

    Article  MathSciNet  Google Scholar 

  31. Kessler, W.S., McPhaden, M.J.: Oceanic equatorial waves and the 1991–1993 El Ni\(\tilde{n}\)o. J. Clim. 8, 1757–1774 (1995)

    Article  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the referees for their careful reading of the manuscript and valuable comments. The authors thank the help from the editor too.

Author information

Authors and Affiliations

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

    Taoyu Yang & JinRong Wang

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

    Michal Fečkan

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

    Michal Fečkan

Authors
  1. Taoyu Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michal Fečkan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. JinRong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to JinRong Wang.

Additional information

Communicated by Adrian Constantin.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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), Guizhou Data Driven Modeling Learning and Optimization Innovation Team ([2020]5016), Major Project of Guizhou Postgraduate Education and Teaching Reform (YJSJGKT[2021]041), the Slovak Research and Development Agency under the contract No. APVV-18-0308, and the Slovak Grant Agency VEGA No.1/0358/20 and No.2/0127/20.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, T., Fečkan, M. & Wang, J. On some azimuthal equatorial flows. Monatsh Math 200, 955–970 (2023). https://doi.org/10.1007/s00605-022-01728-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-022-01728-8

Keywords

Mathematics Subject Classification

Search

Navigation