Ricerche di Matematica

, Volume 68, Issue 2, pp 383–397 | Cite as

Numerical study of the primitive equations in the small viscosity regime

  • F. Gargano
  • M. SammartinoEmail author
  • V. Sciacca


In this paper we study the flow dynamics governed by the primitive equations in the small viscosity regime. We consider an initial setup consisting on two dipolar structures interacting with a no slip boundary at the bottom of the domain. The generated boundary layer is analyzed in terms of the complex singularities of the horizontal pressure gradient and of the vorticity generated at the boundary. The presence of complex singularities is correlated with the appearance of secondary recirculation regions. Two viscosity regimes, with different qualitative properties, can be distinguished in the flow dynamics.


Primitive equations Zero viscosity limit Singularity tracking methods 

Mathematics Subject Classification

35Q35 76F40 35A21 65M70 



The authors thank an anonymous referee for suggestions and comments that helped improving the presentation of the paper. The work of MS and VS was partially supported by GNFM–INdAM. The work of FG was partially supported by the GNFM–INdAM Progetto Giovani Grant.


  1. 1.
    Baker, G., Caflisch, R., Siegel, M.: Singularity formation during Rayleigh–Taylor instability. J. Fluid Mech. 252, 51–75 (1993)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bernardi, C., Maday, Y.: Uniform inf-sup conditions for the spectral discretization of the Stokes problem. Math. Models Methods Appl. Sci. 9(3), 395–414 (1999)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Boyd, J.: Chebyshev and Fourier Spectral Methods, p. 11501. DOVER Publications, Mineoal (2000)Google Scholar
  4. 4.
    Brenier, Y.: Homogeneous hydrostatic flows with convex velocity profiles. Nonlinearity 12(3), 495–512 (1999)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Caflisch, R.: Singularity formation for complex solutions of the 3D incompressible Euler equations. Physica D 67(1–3), 1–18 (1993)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Caflisch, R., Gargano, F., Sammartino, M., Sciacca, V.: Complex singularities and PDEs. Riv. Mat. Univ. Parma 6(1), 69–133 (2015)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Caflisch, R., Gargano, F., Sammartino, M., Sciacca, V.: Regularized Euler-\(\alpha \) motion of an infinite array of vortex sheets. Boll. Unione Mat. Ital. 10(1), 113–141 (2017)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Canuto, C., Hussaini, M., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Computational Physics. Springer, Berlin (1988)zbMATHGoogle Scholar
  9. 9.
    Cao, C., Ibrahim, S., Nakanishi, K., Titi, E.: Finite-time blowup for the inviscid primitive equations of oceanic and atmospheric dynamics. Commun. Math. Phys. 337(2), 473–482 (2015)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Cao, C., Titi, E.: Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics. Ann. Math. 166(1), 245–267 (2007)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Carrier, G., Krook, M., Pearson, C.: Functions of a Complex Variable: Theory and Technique. McGraw-Hill, New York (1966)zbMATHGoogle Scholar
  12. 12.
    Cichowlas, C., Brachet, M.E.: Evolution of complex singularities in Kida–Pelz and Taylor–Green inviscid flows. Fluid Dyn. Res. 36(4–6), 239–248 (2005)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Clercx, H., van Heijst, G.: Dissipation of coherent structures in confined two-dimensional turbulence. Phys. Fluids 29(11), 111,103 (2017)Google Scholar
  14. 14.
    Clercx, H., Bruneau, C.H.: The normal and oblique collision of a dipole with a no-slip boundary. Comput. Fluids 35(3), 245–279 (2006)zbMATHGoogle Scholar
  15. 15.
    Clercx, H., van Heijst, G.: Dissipation of kinetic energy in two-dimensional bounded flows. Phys. Rev. E 65(6), 066,305 (2002)MathSciNetGoogle Scholar
  16. 16.
    Cowley, S.: Computer extension and analytic continuation of Blasius’ expansion for impulsively flow past a circular cylinder. J. Fluid Mech. 135, 389–405 (1983)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Della Rocca, G., Lombardo, M., Sammartino, M., Sciacca, V.: Singularity tracking for Camassa–Holm and Prandtl’s equations. Appl. Numer. Math. 56(8), 1108–1122 (2006)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Frisch, U., Matsumoto, T., Bec, J.: Singularities of Euler flow? Not out of the blue!. J. Stat. Phys. 113(5), 761–781 (2003)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Gargano, F., Ponetti, G., Sammartino, M., Sciacca, V.: Complex singularities in KdV solutions. Ric. Mat. 65(2), 479–490 (2016)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Gargano, F., Sammartino, M., Sciacca, V.: Singularity formation for Prandtl’s equations. Physica D 238(19), 1975–1991 (2009)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Gargano, F., Sammartino, M., Sciacca, V.: High Reynolds number Navier–Stokes solutions and boundary layer separation induced by a rectilinear vortex. Comput. Fluids 52, 73–91 (2011)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Gargano, F., Sammartino, M., Sciacca, V.: Fluid mechanics: singular behavior of a vortex layer in the zero thickness limit. Rend. Lincei Math. Appl. 28(3), 553–572 (2017)zbMATHGoogle Scholar
  23. 23.
    Gargano, F., Sammartino, M., Sciacca, V., Cassel, K.: Analysis of complex singularities in high-Reynolds-number Navier–Stokes solutions. J. Fluid Mech. 747, 381–421 (2014)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Gargano, F., Sammartino, M., Sciacca, V., Cassel, K.: Viscous-inviscid interactions in a boundary-layer flow induced by a vortex array. Acta Appl. Math. 132, 295–305 (2014)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Grenier, E.: On the derivation of homogeneous hydrostatic equations. Math. Model. Numer. Anal. 33(5), 965–970 (1999)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Klein, C., Roidot, K.: Numerical study of shock formation in the dispersionless Kadomtsev–Petviashvili equation and dispersive regularizations. Physica D 265, 1–25 (2013)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Kramer, W., Clercx, H., van Heijst, G.: Vorticity dynamics of a dipole colliding with a no-slip wall. Phys. Fluids 19(12), 126,603 (2007)zbMATHGoogle Scholar
  28. 28.
    Krasny, R.: A study of singularity formation in a vortex sheet by the point-vortex approximation. J. Fluid Mech. 167, 65–93 (1986)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Kukavica, I., Lombardo, M., Sammartino, M.: Zero viscosity limit for analytic solutions of the primitive equations. Arch. Rational Mech. Anal. 222(1), 15–45 (2016)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Kukavica, I., Ziane, M.: On the regularity of the primitive equations of the ocean. Nonlinearity 20(12), 2739–2753 (2007)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Kukavica, I., Ziane, M.: The regularity of solutions of the primitive equations of the ocean in space dimension three. C. R. Math. 345(5), 257–260 (2007)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Levermore, C., Sammartino, M.: A shallow water model with eddy viscosity for basins with varying bottom topography. Nonlinearity 14(6), 1493–1515 (2001)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Lions, J.L., Temam, R., Wang, S.: On the equations of the large-scale ocean. Nonlinearity 5(5), 1007–1053 (1992)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Lions, J.L., Temam, R., Wang, S.: Mathematical theory for the coupled atmosphere–ocean models. J. Math. Pures Appl. 74(2), 105–163 (1995)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Lions, J.L., Temam, R., Wang, S.: A simple global model for the general circulation of the atmosphere. Commun. Pure Appl. Math. 50(8), 707–752 (1997)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Lions, P.L.: Mathematical topics in fluid mechanics, vol. 1. Oxford Lecture Series in Mathematics and its Applications, vol. 3. Incompressible models. The Clarendon Press, Oxford University Press, New York (1996)Google Scholar
  37. 37.
    Masmoudi, N., Wong, T.K.: On the \(H^s\) theory of hydrostatic Euler equations. Arch. Rational Mech. Anal. 204(1), 231–271 (2012)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Matsumoto, T., Bec, J., Frisch, U.: The analytic structure of 2D Euler flow at short times. Fluid Dyn. Res. 36(4–6), 221–237 (2005)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Nguyen Van Yen, R., Farge, M., Schneider, K.: Energy dissipating structures produced by walls in two-dimensional flows at vanishing viscosity. Phys. Rev. Lett. 106, 184502 (2011)Google Scholar
  40. 40.
    Obabko, A., Cassel, K.: Navier-Stokes solutions of unsteady separation induced by a vortex. J. Fluid Mech. 465, 99–130 (2002)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Obabko, A., Cassel, K.: On the ejection-induced instability in Navier–Stokes solutions of unsteady separation. Philos. Trans. A Math. Phys. Eng. Sci. 363(1830), 1189–1198 (2005)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Pauls, W., Matsumoto, T., Frisch, U., Bec, J.: Nature of complex singularities for the 2D Euler equation. Physica D 219(1), 40–59 (2006)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Pedlosky, J.: Geophysical Fluid Dynamics. Springer, New York (1979)zbMATHGoogle Scholar
  44. 44.
    Peyret, R.: Spectral Methods for Incompressible Viscous Flow. Springer, New York (2002)zbMATHGoogle Scholar
  45. 45.
    Schonbek, M.: \(l^2\) decay for weak solutions of the Navier–Stokes equations. Arch. Rational Mech. Anal. 88(3), 209–222 (1985)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Sciacca, V., Schonbek, M., Sammartino, M.: Long time behavior for a dissipative shallow water model. Ann. Inst. H. Poincare Anal 34(3), 731–757 (2017)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Shelley, M.: A study of singularity formation in vortex-sheet motion by a spectrally accurate vortex method. J. Fluid. Mech. 244, 493–526 (1992)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Sohn, S.I.: Singularity formation and nonlinear evolution of a viscous vortex sheet model. Phys. Fluids 25(1), 014,106 (2013)Google Scholar
  49. 49.
    Temam, R., Ziane, M.: Some Mathematical Problems in Geophysical Fluid Dynamics. Handbook of Mathematical Fluid Dynamics, vol. III. North-Holland, Amsterdam (2004)zbMATHGoogle Scholar
  50. 50.
    Trefethen, L.: Spectral Methods in MATLAB. Environments, and Tools. Society for Industrial and Applied Mathematics, Software (2000)Google Scholar
  51. 51.
    Zheng, L., Wang, S.: Finite-time blowup for the 3-D primitive equations of oceanic and atmospheric dynamics. Appl. Math. Lett. 76, 117–122 (2018)MathSciNetzbMATHGoogle Scholar

Copyright information

© Università degli Studi di Napoli "Federico II" 2018

Authors and Affiliations

  1. 1.Dipartimento di Energia, Ingegneria dell’Informazione e Modelli Matematici (DEIM)Università degli Studi di PalermoPalermoItaly
  2. 2.Dipartimento dell’Innovazione Industriale e Digitale (DIID)Università degli Studi di PalermoPalermoItaly
  3. 3.Dipartimento di Matematica e InformaticaUniversità degli Studi di PalermoPalermoItaly

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