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Archive for Rational Mechanics and Analysis

, Volume 231, Issue 3, pp 1853–1915 | Cite as

Imperfect Bifurcation for the Quasi-Geostrophic Shallow-Water Equations

  • David Gerard Dritschel
  • Taoufik HmidiEmail author
  • Coralie Renault
Article

Abstract

We study analytical and numerical aspects of the bifurcation diagram of simply connected rotating vortex patch equilibria for the quasi-geostrophic shallow-water (QGSW) equations. The QGSW equations are a generalisation of the Euler equations and contain an additional parameter, the Rossby deformation length \({\varepsilon^{-1}}\), which enters into the relation between the stream function and (potential) vorticity. The Euler equations are recovered in the limit \({\varepsilon \rightarrow 0}\). We prove, close to circular (Rankine) vortices, the persistence of the bifurcation diagram for arbitrary Rossby deformation length. However we show that the two-fold branch, corresponding to Kirchhoff ellipses for the Euler equations, is never connected even for small values \({\varepsilon}\), and indeed is split into a countable set of disjoint connected branches. Accurate numerical calculations of the global structure of the bifurcation diagram and of the limiting equilibrium states are also presented to complement the mathematical analysis.

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Notes

Acknowledgments

DGD received support for this research from the UK Engineering and Physical Sciences Research Council (grant number EP/H001794/1). TH is partially supported by the the ANR project Dyficolti ANR-13-BS01-0003- 01.

References

  1. 1.
    Burbea J.: Motions of vortex patches. Lett. Math. Phys. 6(1), 1–16 (1982)ADSMathSciNetzbMATHGoogle Scholar
  2. 2.
    Castro A., Córdoba D., Gomez-Serrano J.: Existence and regularity of rotating global solutions for the generalized surface quasi-geostrophic equations. Duke Math. J. 165(5), 935–984 (2016)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Castro, A., Córdoba, D., Gomez-Serrano, J.: Uniformly rotating analytic global patch solutions for active scalars. Ann. PDE 2(1), Art. 1, 34 2016Google Scholar
  4. 4.
    Cerretelli C., WilliamsonC. H. K.: A new family of uniform vortices related to vortex configurations before merging. J. Fluid Mech. 493, 219–229 (2003)ADSMathSciNetzbMATHGoogle Scholar
  5. 5.
    Crandall M.G., Rabinowitz P.H.: Bifurcation from simple eigenvalues. J.Funct. Analysis 8, 321–340 (1971)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Deem G.S., Zabusky N.J.: Vortex waves: stationary "V-states", interactions, recurrence, and breaking. Phys. Rev. Lett. 40(13), 859–862 (1978)ADSGoogle Scholar
  7. 7.
    Dritschel D.G.: The nonlinear evolution of rotating configurations of uniform vorticity. J. Fluid Mech. 172, 157–172 (1986)ADSzbMATHGoogle Scholar
  8. 8.
    Dritschel D.G.: Contour surgery: a topological reconnection scheme for extended integrations using contour dynamics. J. Comput. Phys. 77, 240–266 (1988)ADSMathSciNetzbMATHGoogle Scholar
  9. 9.
    Dritschel D.G.: Contour dynamics and contour surgery: numerical algorithms for extended, high-resolution modelling of vortex dynamics in two-dimensional, inviscid, incompressible flows. Computer Phys. Rep. 10, 77–146 (1989)ADSGoogle Scholar
  10. 10.
    Dritschel D.G.: A general theory for two-dimensional vortex interactions. J. Fluid Mech. 293, 269–303 (1995)ADSMathSciNetzbMATHGoogle Scholar
  11. 11.
    Golubitsky M., Schaeffer D.: A theory for imperfect bifurcation via singularity theory. Commun. Pure Appl. Math. 32(1), 21–98 (1979)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hassainia Z., Hmidi T.: On the V-states for the generalized quasi-geostrophic equations. Commun. Math. Phys. 337(1), 321–377 (2015)ADSMathSciNetzbMATHGoogle Scholar
  13. 13.
    Hassainia, Z., Masmoudi, N.,Wheeler, M. H.: Global bifurcation of rotating vortex patches, arXiv:1712.03085
  14. 14.
    Hmidi T., Mateu J., Verdera J.: Boundary regularity of rotating vortex patches. Arch. Ration. Mech. Anal. 209(1), 171–208 (2013)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Hmidi T., Mateu J.: Bifurcation of rotating patches from Kirchhoff vortices. Discrete Contin. Dyn. Syst. 36(10), 5401–5422 (2016)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Kamm, J. R.: Shape and stability of two-dimensional uniform vorticity regions. Ph.D. thesis, California Institute of Technology, 1987Google Scholar
  17. 17.
    Kirchhoff G.R.: Vorlesungenb̈er mathematische Physik.Mechanik. Teubner, Leipzig (1876)Google Scholar
  18. 18.
    Liu P., Shi J., Wang Y.: Imperfect transcritical and pitchfork bifurcations. J. Funct. Anal. 251(2), 573–600 (2007)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Love A.E.H.: On the stability of certain vortex motions. Proc. Lond. Math. Soc. 35, 18 (1893)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Luzzatto-Fegiz P., Williamson C.H.K.: Stability of elliptical vortices from “Imperfect–Velocity–Impulse” diagrams. Theor. Comput. Fluid Dyn. 24(1-4), 181–188 (2010)zbMATHGoogle Scholar
  21. 21.
    Luzzatto-Fegiz P., Williamson C.H.K.: An efficient and general numerical method to compute steady uniform vortices. J. Comput. Phys. 230, 6495–6511 (2011)ADSMathSciNetzbMATHGoogle Scholar
  22. 22.
    Overman E.A. II: Steady-state solutions of the Euler Equations in two dimensions II Local analysis of limiting V-states.. SIAM J. Appl. Math. 46(5), 765–800 (1986)ADSMathSciNetzbMATHGoogle Scholar
  23. 23.
    PŁotka H., Dritschel D.G.: Quasi-geostrophic shallow-water vortex-patch equilibria and their stability. Geophys. Astrophys. Fluid Dyn. 106(6), 574–595 (2012)ADSMathSciNetGoogle Scholar
  24. 24.
    Polvani, L. M.: Geostrophic vortex dynamics. PhD thesis, MIT/WHOI WHOI-88-48, 1988Google Scholar
  25. 25.
    Polvani L.M., Zabusky N.J., Flierl G.R.: Two-layer geostrophic vortex dynamics. Part 1. Upper-layer V-states and merger. J. Fluid Mech. 205, 215–242 (1989)ADSMathSciNetzbMATHGoogle Scholar
  26. 26.
    Segura J.: Bounds for ratios of modified Bessel functions and associated Turán-type inequalities. J. Math. Anal. Appl. 374, 516–528 (2011)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Shi J.: Persistence and bifurcation of degenerate solutions. J. Funct. Anal. 169(2), 494–531 (1999)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Vallis G.K.: Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press, Cambridge (2008)zbMATHGoogle Scholar
  29. 29.
    Watson G.N.: A Treatise on the Theory of Bessel Functions. Cambrige University Press, Cambridge (1944)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • David Gerard Dritschel
    • 1
  • Taoufik Hmidi
    • 2
    Email author
  • Coralie Renault
    • 2
  1. 1.Mathematical InstituteUniversity of St AndrewsSt AndrewsUK
  2. 2.CNRS, IRMAR - UMR 6625, Univ RennesRennesFrance

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