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On Some Azimuthally Propagating Flows with Stratification

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Abstract

We present an exact solution to the governing equations for some azimuthally propagating flows with stratification in the spherical coordinates. Using the short-wavelength perturbation method in the spirit of Constantin, Germain (J Geophys Res Oceans 118:2802-2810, 2013), Henry, Martin (J Diff Equ 266:6788-6808, 2019), Henry, Martin (Arch Rat Mech Anal 233:497-512, 2019), Ionescu-Kruse, Martin (J Math Fluid Mech 20:7-34, 2018), we prove that such flows are linearly stable for a specific choice of the density.

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References

  1. Bayly, B.J.: Three-dimensional instabilities in qusi-two-dimensional inviscid flows, Nonlinear Wave Interactions in Fluids, 71-77, (1987)

  2. 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 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  5. Constantin, A., Germain, P.: Instability of some equatorially trapped waves. J. Geophys. Res.: Oceans 118, 2802–2810 (2013)

    Article  Google Scholar 

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

    Article  Google Scholar 

  7. 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 

  8. 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 

  9. 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  Google Scholar 

  10. 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. Fluid 29, 056604 (2017)

    Article  Google Scholar 

  11. Constantin, A., Johnson, R.S.: Steady large-scale ocean flows in spherical coordinates. Oceanography 31, 42–50 (2018)

    Article  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)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Fan, L., Gao, H.: Instability of equatorial edge waves in the background flow. Proceed. Am. Math. Soci. 145, 765–778 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Friedlander, S., Vishik, M.M.: Instability criteria for the flow of an inviscid incompressible fluid. Phys. Rev. Lett. 66, 2204–2206 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gerstner, F.: Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile. Annalen Der Physik 2, 412–445 (1809)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Henry, D.: On three-dimensional Gerstner-like equatorial water waves. Philos. Trans. Royal Soci. A: Math., Phys. Eng. Sci. 376, 20170088 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Henry, D., Lyons, T.: Pollard waves with underlying currents. Proceed. Am. Math. Soc. 149, 1175–1188 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ionescu-Kruse, D.: Short-wavelength instabilities of edge waves in stratified water. Discr. Contin. Dyn. Syst. 35, 2053–2066 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ionescu-Kruse, D., Martin, C.I.: Local stability for an exact steady purely azimuthal equatorial flow. J. Math. Fluid Mech. 20, 7–34 (2018)

    MathSciNet  MATH  Google Scholar 

  26. Johnson, R.S.: The ocean and the atmosphere: an applied mathematician’s view. Commun. Pure Appl. Anal. 21, 2357–2381 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kluczek, M.: Nonhydrostatic Pollard-like internal geophysical waves. Discr. Contin. Dyn. Sys. 39, 5171–5183 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  28. Leblanc, S.: Local stability of Gerstner’s waves. J. Fluid Mech. 506, 245–254 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lifschitz, A., Hameiri, E.: Local stability conditions in fluid dynamics. Phys. Fluid. 3, 2644–2651 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  30. 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 

  31. Miao, F., Fečkan, M., Wang, J.: Exact solution and instability for geophysical edge waves. Commun. Pure Appl. Anal. 21, 2447–2461 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  32. Miao, F., Fečkan, M., Wang, J.: Stratified equatorial flows in the \(\beta \)-plane approximation with a free surface. Monatsh. Math. 200, 315–334 (2023)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  34. Stuhlmeier, R.: On edge waves in stratified water along a sloping beach. J. Nonlin. Math. Phys. 18, 127–137 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  36. Wang, J., Fečkan, M., Guan, Y.: Local and global analysis for discontinuous atmospheric Ekman equations. J. Dyn. Diff. Equ. 35, 663–677 (2023)

    Article  MathSciNet  Google Scholar 

  37. Wunsch, C.: Modern observational physical oceanography: understanding the global ocean. Princeton University Press, Princeton (2015)

    Google Scholar 

  38. Yang, T., Fečkan, M., Wang, J.: On some azimuthal equatorial flows. Monatsh. Math. 200, 955–970 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhang, W., Fečkan, M., Wang, J.: Positive solutions to integral boundary value problems from geophysical fluid flows. Monatsh. Math. 193, 901–925 (2020)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

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

    Fahe Miao & JinRong Wang

  2. Supercomputing Algorithm and Application Laboratory of Guizhou University and Gui’an Scientific Innovation Company, Guizhou University, Guiyang, 550025, Guizhou, China

    Fahe Miao & 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 Fećkan

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

    Michal Fećkan

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  1. Fahe Miao
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  2. Michal Fećkan
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Correspondence to JinRong Wang.

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This work is partially supported by the National Natural Science Foundation of China (12161015), Guizhou Provincial Basic Research Program (Natural Science) [2023]034, Qian Ke He Ping Tai Ren Cai-YSZ[2022]002, the Slovak Research and Development Agency under the contract No. APVV-18-0308, and the Slovak Grant Agency VEGA No. 2/0127/20 and No. 1/0084/23.

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Miao, F., Fećkan, M. & Wang, J. On Some Azimuthally Propagating Flows with Stratification. Qual. Theory Dyn. Syst. 22, 85 (2023). https://doi.org/10.1007/s12346-023-00783-2

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