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A Weighted Sturm–Liouville Problem Related to Ocean Flows

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Abstract

We discuss some existence and uniqueness results for the solutions to a two-point boundary-value problem that models the flow of the Antarctic Circumpolar Current in rotating spherical coordinates.

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References

  1. 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)

    Article  ADS  Google Scholar 

  2. Constantin, A., Monismith, S.G.: Gerstner waves in the presence of mean currents and rotation. J. Fluid Mech. 820, 511–528 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Walton, D.W.H.: Antarctica: Global Science from a Frozen Continent. Cambridge University Press, Cambridge (2013)

    Book  Google Scholar 

  4. Chu, J.: On a differential equation arising in geophysics. Monatsh Math. (2017). https://doi.org/10.1007/s00605-017-1087-1

    Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  6. Marynets, K.: On a two-point boundary-value problem in geophysics, Appl. Anal. https://doi.org/10.1080/00036811.2017.1395869

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  9. Jonsson, I.G.: Wave-current interactions. In: Le Méhauté, B., Hanes, D.M. (eds.) The Sea. Ocean Eng. Sc., vol. 9(A), pp. 65–120. Wiley, Hoboken (1990)

    Google Scholar 

  10. Constantin, A., Strauss, W., Varvaruca, E.: Global bifurcation of steady gravity water waves with critical layers. Acta Math. 217, 195–262 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ewing, J.A.: Wind, wave and current data for the design of ships and offshore structures. Mar. Struct. 3, 421–459 (1990)

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  13. Constantin, A.: Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 81. SIAM, Philadelphia (2011)

    Book  Google Scholar 

  14. Henry, D.: Large amplitude steady periodic waves for fixed-depth rotational flows. Commun. Partial Differ. Equ. 38, 1015–1037 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Thomas, G.P.: Wave-current interactions: an experimental and numerical study. J. Fluid Mech. 216, 505–536 (1990)

    Article  ADS  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  17. Constantin, A.: Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves. J. Phys. Oceanogr. 44, 781–789 (2014)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  19. Henry, D.: An exact solution for equatorial geophysical water waves with an underlying current. Eur. J. Mech. B/Fluids 38, 18–21 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Henry, D.: Equatorially trapped nonlinear water waves in a-plane approximation with centripetal forces. J. Fluid Mech. 804, R1 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Henry, D., Sastre-Gomez, S.: Mean flow velocities and mass transport for equatorially-trapped water waves with an underlying current. J. Math. Fluid Mech. 18, 795–804 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Sanjurjo, A.R., Kluczek, M.: Mean flow properties for equatorially trapped internal water wave-current interactions. Appl. Anal. 96, 2333–2345 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Stakgold, I., Holst, M.: Green’s Functions and Boundary Value Problems. Wiley, Hoboken (2011)

    Book  MATH  Google Scholar 

  24. Richardson, R.: Contributions to the study of oscillation properties of the solutions of linear differential equations of the second order. Am. J. Math. 60, 283–316 (1918)

    Article  MathSciNet  Google Scholar 

  25. Atkinson, F.V., Mingarelli, A.B.: Asymptotics of the number of zeros and of the eigenvalues of general-weighted Sturm-Liouville problems. J. Reine Angew. Math. 375(376), 380–393 (1987)

    MathSciNet  MATH  Google Scholar 

  26. Constantin, A.: A general-weighted Sturm-Liouville problem. Annali della Scuola Normale, serie 4, 24(4), 767–782 (1997)

  27. Magnus, W., Winkler, S.: Hill’s Equation. Interscience Publ, New York (1966)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Uzhhorod National University, Universytetska St. 14, Uzhhorod, 88000, Ukraine

    Kateryna Marynets

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  1. Kateryna Marynets
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Correspondence to Kateryna Marynets.

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Communicated by A. Constantin

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Marynets, K. A Weighted Sturm–Liouville Problem Related to Ocean Flows. J. Math. Fluid Mech. 20, 929–935 (2018). https://doi.org/10.1007/s00021-017-0347-0

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  • DOI: https://doi.org/10.1007/s00021-017-0347-0

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