On three-dimensional geophysical capillary–gravity water flows with constant vorticity


Consideration in this paper is three-dimensional capillary–gravity water flows governed by the geophysical water wave equations with all the Coriolis terms being retained. It is proved that the merely possible flow exhibiting a constant vorticity vector captures vanishing vertical velocity, constant horizontal velocity and flat free surface.

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


  1. 1.

    Basu, B., Martin, C.I.: Resonant interactions of rotational water waves in the equatorial \(f\)-plane approximation. J. Math. Phys. 59, 103101 (2018)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Chu, J., Escher, J.: Steady equatorial water waves with vorticity. Discret. Contin. Dyn. Syst. 39, 4713–4729 (2019)

    MathSciNet  MATH  Google Scholar 

  3. 3.

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

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Chu, J., Yang, Y.: Constant vorticity geophysical waves with centripetal forces and at arbitrary latitude (2019). arXiv preprint arXiv:1910.02360

  5. 5.

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

    Google Scholar 

  6. 6.

    Constantin, A.: Two-dimensionality of gravity water flows of constant non-zero vorticity beneath a surface wave train. Eur. J. Mech. B/Fluids 30, 12–16 (2011)

    MathSciNet  MATH  Google Scholar 

  7. 7.

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

    Google Scholar 

  8. 8.

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

    Google Scholar 

  9. 9.

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

    Google Scholar 

  10. 10.

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

    Google Scholar 

  11. 11.

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

    Google Scholar 

  12. 12.

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

    MathSciNet  MATH  Google Scholar 

  13. 13.

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

    Google Scholar 

  14. 14.

    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)

    Google Scholar 

  15. 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)

    Google Scholar 

  16. 16.

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

    Google Scholar 

  17. 17.

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

    Google Scholar 

  18. 18.

    Constantin, A., Kartashova, E.: Effect of non-zero constant vorticity on the nonlinear resonances of capillary water waves. Europhys. Lett. 86, 29001 (2009)

    Google Scholar 

  19. 19.

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

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Craig, W.: Non-existence of solitary water waves in three dimensions. R. Soc. Lond. Philos. Trans. Ser. A: Math. Phys. Eng. Sci. 360, 2127–2135 (2002)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Craig, W., Nicholls, D.P.: Travelling two and three dimensional capillary gravity water waves. SIAM J. Math. Anal. 32, 323–359 (2000)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Cushman-Roisin, B., Beckers, J.-M.: Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, vol. 101. Academic Press, New York (2011)

    Google Scholar 

  23. 23.

    Fan, L., Gao, H., xiao, Q.: An exact solution for geophysical trapped waves in the presence of an underlying current. Dyn. Partial Differ. Equ. 15, 201–214 (2018)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Gallagher, I., Saint-Raymond, L.: On the influence of the Earth’s rotation on geophysical flows. In: Handbook of Mathematical Fluid Mechanics, vol. 4, pp. 201–329 (2007)

  25. 25.

    Groves, M.D., Haragus, M.: A bifurcation theory for three-dimensional oblique travelling gravity-capillary water waves. J. Nonlinear Sci. 13, 397–447 (2003)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Henry, D.: Internal equatorial water waves in the \(f\)-plane. J. Nonlinear Math. Phys. 22, 499–506 (2015)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Henry, D.: Exact equatorial water waves in the \(f\)-plane. Nonlinear Anal.: Real World Appl. 28, 284–289 (2016)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Henry, D.: Equatorially trapped nonlinear water waves in a \(\beta \)-plane approximation with centriptal forces. J. Fluid Mech. 804 (2016). https://doi.org/10.1017/jfm.2016.544

  29. 29.

    Henry, D.: A modified equatorial \(\beta \)-plane approximation modelling nonlinear wave-current interactions. J. Differ. Equ. 263, 2554–2566 (2017)

    MathSciNet  MATH  Google Scholar 

  30. 30.

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

    MathSciNet  MATH  Google Scholar 

  31. 31.

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

    MathSciNet  MATH  Google Scholar 

  32. 32.

    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 

  33. 33.

    Ionescu-Kruse, D.: Exponential profiles producing genuine three-dimensional nonlinear flows relevant for equatorial ocean dynamics. J. Differ. Equ. 268, 1326–1344 (2020)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Iooss, G., Plotnikov, P.I.: Small divisor problem in the theory of three-dimensional water gravity waves. Mem. Am. Math. Soc. 200, 940 (2009)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Iooss, G., Plotnikov, P.I.: Asymmetrical three-dimensional travelling gravity waves. Arch. Ration. Mech. Anal. 200, 789–880 (2011)

    MathSciNet  MATH  Google Scholar 

  36. 36.

    Lighthill, J.: Waves in Fluids. Cambridge University Press, Cambridge (1978)

    Google Scholar 

  37. 37.

    Martin, C.I.: Non-existence of time-dependent three-dimensional gravity water flows with constant non-zero vorticity. Physics of Fluids 30, 107102 (2018)

    Google Scholar 

  38. 38.

    Martin, C.I.: Resonant interactions of capillary-gravity water waves. J. Nonlinear Math. Phys. 19, 807–817 (2017)

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Martin, C.I.: Two-dimensionality of gravity water flows governed by the equatorial \(f\)-plane approximation. Annali di Matematica Pura ed Applicata 196, 2253–2260 (2017)

    MathSciNet  MATH  Google Scholar 

  40. 40.

    Martin, C.I.: On the vorticity of mesoscale ocean currents. Oceanography 31, 28–35 (2018)

    Google Scholar 

  41. 41.

    Martin, C.I.: On constant vorticity water flows in the \(\beta \)-plane approximation. J. Fluid Mech. 865, 762–774 (2019)

    MathSciNet  MATH  Google Scholar 

  42. 42.

    Martin, C.I.: Constant vorticity water flows with full Coriolis term. Nonlinearity 32, 2327–2336 (2019)

    MathSciNet  MATH  Google Scholar 

  43. 43.

    Perlin, M., Schultz, W.W.: Capillary effects on surface waves. Annu. Rev. Fluid Mech. 32, 241–274 (2000)

    MathSciNet  MATH  Google Scholar 

  44. 44.

    Sajjadi, S.: Vorticity generated by pure capillary waves. J. Fluid Mech. 459, 277–288 (2002)

    MathSciNet  MATH  Google Scholar 

  45. 45.

    Reeder, J., Shinbrot, M.: Three-dimensional, nonlinear wave interaction in water of constant depth. Nonlinear Anal.: Theory Methods Appl. 5, 303–323 (1981)

    MathSciNet  MATH  Google Scholar 

  46. 46.

    Stuhlmeier, R.: On constant vorticity flows beneath two-dimensional surface solitary waves. J. Nonlinear Math. Phys. 19, 1240004 (2012)

    MathSciNet  MATH  Google Scholar 

  47. 47.

    Vallis, G.K.: Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press, Cambridge (2006)

    Google Scholar 

  48. 48.

    Wahlén, E.: Non-existence of three-dimensional travelling water waves with constant non-zero vorticity. J. Fluid Mech. 746 (2014). https://doi.org/10.1017/jfm.2014.131

  49. 49.

    Wheeler, M.H.: Simplified models for equatorial waves with vertical structure. Oceanography 31, 36–41 (2018)

    Google Scholar 

Download references


The work of Fan is supported by a NSFC Grant No. 11701155. The work of Gao is partially supported by the NSFC Grant No. 11531006 and Jiangsu Center for Collaborative Innovation in Geographical Information Resource Development and Application. The authors are grateful to Calin Iulian Martin for his valuable suggestions during preparation of the manuscript.

Author information



Corresponding author

Correspondence to Hongjun Gao.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Fan, L., Gao, H. On three-dimensional geophysical capillary–gravity water flows with constant vorticity. Annali di Matematica (2020). https://doi.org/10.1007/s10231-020-01010-4

Download citation


  • Capillary–gravity waves
  • Coriolis force
  • Waves in rotating fluids
  • Vorticity

Mathematics Subject Classification

  • 35Q35
  • 76B45
  • 76U05