Skip to main content
SpringerLink
Log in

Multiple Solutions for an Elliptic Equation from the Antarctic Circumpolar Current

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

In this paper, a mathematical model of the second order elliptic equation of the Antarctic Circumpolar Current with Dirichlet boundary is established. By introducing truncation function and perturbation method, the existence of infinitely many solutions for the nonlinear elliptic equations is obtained when the nonlinear ocean vorticity is subcritical growth and super-quadratic.

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

Similar content being viewed by others

References

  1. Zhong, W., Zhao, J., Shi, J., Cao, Y.: The Beaufort Gyre variation and its impacts on the Canada Basin in 2003–2012. Acta Oceanol. Sin. 34, 19–31 (2015)

    Google Scholar 

  2. Constantin, A., Johnson, R.S.: Large gyres as a shallow-water asymptotic solution of Euler’s equation in spherical coordinates. Proc. R. Soc. A Math. Phys. Eng. Sci. 473, 20170063 (2017)

    MathSciNet  MATH  Google Scholar 

  3. Viudez, A., Dritschel, D.G.: Vertical velocity in mesoscale geophysical flows. J. Fluid Mech. 483, 199–223 (2015)

    MathSciNet  MATH  Google Scholar 

  4. Danabasoglu, G., McWilliams, J.C., Gent, P.R.: The role of mesoscale tracer transport in the global ocean circulation. Science 264, 1123–1126 (1994)

    Google Scholar 

  5. Farneti, R., Delworth, T.L., Rosati, A., Griffies, S.M., Zeng, F.: The role of mesoscale eddies in the rectification of the Southern Ocean response to climate change. J. Phys. Oceanogr. 40, 1539–1557 (2010)

    Google Scholar 

  6. Firing, Y.L., Chereskin, T.K., Mazloff, M.R.: Vertical structure and transport of the Antarctic Circumpolar Current in Drake Passage from direct velocity observations. J. Geophys. Res. Atmos. 116, C08015 (2011)

    Google Scholar 

  7. Ivchenko, V.O., Richards, K.J.: The dynamics of the Antarctic circumpolar current. J. Phys. Oceanogr. 26, 753–774 (2012)

    Google Scholar 

  8. Klinck, J., Nowlin, W.D.: Antarctic Circumpolar Current. Academic Press, Cambridge (2001)

    Google Scholar 

  9. Marshall, D.P., Munday, D.R., Allison, L.C., Hay, R.J., Johnson, H.L.: Gill’s model of the Antarctic Circumpolar Current, revisited: the role of latitudinal variations in wind stress. Ocean Model. 97, 37–51 (2016)

    Google Scholar 

  10. Thompson, A.F.: The atmospheric ocean: eddies and jets in the Antarctic Circumpolar Current. Philos. Trans. R. Soc. A-Math. Phys. Eng. Sci. 366, 4529–4541 (2008)

    MathSciNet  Google Scholar 

  11. Wolff, J.O.: Modelling the Antarctic Circumpolar Current: eddy-dynamics and their parametrization. Environ. Model. Softw. 14, 317–326 (1999)

    Google Scholar 

  12. Constantin, A., Johnson, R.S.: Large-scale oceanic currents as shallow water asymptotic solutions of the Navier–Stokes equation in rotating spherical coordinates. Deep-Sea Res. Part II Top Stud. Oceanogr. 160, 32–40 (2019)

    Google Scholar 

  13. Constantin, A., Johnson, R.S.: Ekman-type solutions for shallow-water flows on a rotating sphere: a new perspective on a classical problem. Phys. Fluids 31, 021401 (2019)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  16. 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)

    MathSciNet  MATH  Google Scholar 

  17. Matioc, A.V.: An exact solution for geophysical equatorial edge waves over a sloping beach. J. Phys. A Math. Theor. 45, 365501 (2012)

    MathSciNet  MATH  Google Scholar 

  18. Matioc, A.V.: An explicit solution for deep water waves with coriolis effects. J. Nonlinear Math. Phys. 19, 1240005 (2012)

    MathSciNet  MATH  Google Scholar 

  19. Ionescu-Kruse, D.: An exact solution for geophysical edge waves in the \(f\)-plane approximation. Nonlinear Anal. Real World Appl. 24, 190–195 (2015)

    MathSciNet  MATH  Google Scholar 

  20. Miao, F., Fečkan, M., Wang, J.: A new approach to study constant vorticity water flows in the \(\beta \)-plane approximation with centripetal forces, Dynamics of Partial. Differ. Equ. 18, 199–210 (2021)

    MathSciNet  MATH  Google Scholar 

  21. Marynets, K.: A nonlinear two-point boundary-value problem in geophysics. Monatshefte für Mathematik 188, 287–295 (2019)

    MathSciNet  MATH  Google Scholar 

  22. Marynets, K.: On a two-point boundary-value problem in geophysics. Appl. Anal. 98, 553–560 (2019)

    MathSciNet  MATH  Google Scholar 

  23. Marynets, K.: A weighted Sturm-Liouville problem related to ocean flows. J. Math. Fluid Mech. 20, 929–935 (2018)

    MathSciNet  MATH  Google Scholar 

  24. Chu, J.: On a differential equation arising in geophysics. Monatshefte für Mathematik 187, 499–508 (2018)

    MathSciNet  MATH  Google Scholar 

  25. Chu, J.: On a nonlinear model for arctic gyres. Ann. Mat. 197, 651–659 (2018)

    MathSciNet  MATH  Google Scholar 

  26. Chu, J.: Monotone solutions of a nonlinear differential equation for geophysical fluid flows, Nonlinear. Analysis 166, 144–153 (2018)

    MathSciNet  MATH  Google Scholar 

  27. Chu, J.: On an infinite-interval boundary-value problem in geophysics. Monatshefte für Mathematik 188, 621–628 (2019)

    MathSciNet  MATH  Google Scholar 

  28. Chu, J., Marynets, K.: Nonlinear differential equations modeling the Antarctic Circumpolar Current. J. Math. Fluid Mech. 23, 92 (2021)

    MathSciNet  MATH  Google Scholar 

  29. Martin, C.I., Quirchmayr, R.: A steady stratied purely azimuthal flow representing the Antarctic Circumpolar Current. Monatshefte für Mathematik 192, 401–407 (2020)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  32. Wang, J., Fečkan, M., Zhang, W.: On the nonlocal boundary value problem of geophysical fluid flows. Z. Angew. Math. Phys. 72, 27 (2021)

    MathSciNet  MATH  Google Scholar 

  33. Wang, J., Fečkan, M., Wen, Q., O’Regan, D.: Existence and uniqueness results for modeling jet flow of the Antarctic Circumpolar Current. Monatshefte für Mathematik 194, 1–21 (2021)

    MathSciNet  MATH  Google Scholar 

  34. Wang, J., Zhang, W., Fečkan, M.: Periodic boundary value problem for second-order differential equations from geophysical fluid flows. Monatshefte für Mathematik 195, 523–540 (2021)

    MathSciNet  MATH  Google Scholar 

  35. Fečkan, M., Li, Q., Wang, J.: Existence and Ulam-Hyers stability of positive solutions for a nonlinear model for the Antarctic Circumpolar Current. Monatshefte für Mathematik 197, 419–434 (2022)

    MathSciNet  MATH  Google Scholar 

  36. Zhang, W., Wang, J., Fečkan, M.: Existence and uniqueness results for a second order differential equation for the ocean flow in arctic gyres. Monatshefte für Mathematik 193, 177–192 (2020)

    MathSciNet  MATH  Google Scholar 

  37. Zhang, W., Fečkan, M., Wang, J.: Positive solutions to integral boundary value problems from geophysical fluid flows. Monatshefte für Mathematik 193, 901–925 (2020)

    MathSciNet  MATH  Google Scholar 

  38. Haziot, S.V.: Explicit two-dimensional solutions for the ocean flow in arctic gyres. Monatshefte für Mathematik 189, 429–440 (2019)

    MathSciNet  MATH  Google Scholar 

  39. Haziot, S.V.: Study of an elliptic partial differential equation modelling the Antarctic Circumpolar Current. Discrete Contin. Dyn. Syst. 39, 4415–4427 (2019)

    MathSciNet  MATH  Google Scholar 

  40. Fečkan, M., Wang, J., Zhang, W.: Existence of solutions for nonlinear elliptic equations modeling the steady flow of the Antarctic Circumpolar Current. Differ. Integr. Equ. 35, 277–298 (2022)

    MathSciNet  MATH  Google Scholar 

  41. Daners, D.: The Mercator and stereographic projections, and many in between. Am. Math. Monthly 119, 199–210 (2012)

    MathSciNet  MATH  Google Scholar 

  42. Clark, D.C.: A variant of the Ljusternick-Schnirelman theory. Indiana Univ. Math. J. 22, 65–74 (1972)

    MathSciNet  MATH  Google Scholar 

  43. Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. I. Interscience, New York (1953)

    MATH  Google Scholar 

  44. Agmon, S.: Lectures on Elliptic Boundary Value Problems, Van Nostrand Company Inc., Princeton N. J., (1965)

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. School of Mathematics and Statistics, Liupanshui Normal University, Liupanshui, 553004, Guizhou, China

    WenLin Zhang

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

    WenLin Zhang & JinRong Wang

  3. 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

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

    Michal Fečkan

Authors
  1. WenLin Zhang
    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

Contributions

WZ wrote the main manuscript text, MF and JW checked and corrected the manuscript.

Corresponding author

Correspondence to JinRong Wang.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Publisher's Note

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

This work is partially supported by Scientific Research Project of Liupanshui Normal University (LPSSYYBZK202209), Guizhou Provincial Science and Technology Projects (ZK[2022]535), Qian Ke He Ping Tai Ren Cai-YSZ[2022]002, the Slovak Research and Development Agency under the contract No. APVV-18-0308 and by the Slovak Grant Agency VEGA No. 2/0127/20 and No. 1/0084/23.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, W., Fečkan, M. & Wang, J. Multiple Solutions for an Elliptic Equation from the Antarctic Circumpolar Current. Qual. Theory Dyn. Syst. 22, 45 (2023). https://doi.org/10.1007/s12346-023-00751-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-023-00751-w

Keywords

Search

Navigation