Advertisement

Communications in Mathematical Physics

, Volume 350, Issue 2, pp 699–747 | Cite as

Existence of Corotating and Counter-Rotating Vortex Pairs for Active Scalar Equations

  • Taoufik HmidiEmail author
  • Joan Mateu
Article

Abstract

In this paper, we study the existence of corotating and counter-rotating pairs of simply connected patches for Euler equations and the \({{\rm (SQG)}_{\alpha}}\) equations with \({\alpha \in (0,1)}\). From the numerical experiments implemented for Euler equations in Deem and Zabusky (Phys Rev Lett 40(13):859–862, 1978), Pierrehumbert (J Fluid Mech 99:129–144, 1980), Saffman and Szeto (Phys Fluids 23(12):2339–2342, 1980) it is conjectured the existence of a curve of steady vortex pairs passing through the point vortex pairs. There are some analytical proofs based on variational principle (Keady in J Aust Math Soc Ser B 26:487–502, 1985; Turkington in Nonlinear Anal Theory Methods Appl 9(4):351–369, 1985); however, they do not give enough information about the pairs, such as the uniqueness or the topological structure of each single vortex. We intend in this paper to give direct proofs confirming the numerical experiments and extend these results for the \({{\rm (SQG)}_{\alpha}}\) equation when \({\alpha \in (0,1)}\). The proofs rely on the contour dynamics equations combined with a desingularization of the point vortex pairs and the application of the implicit function theorem.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aref H.: Integrable, chaotic, and turbulent vortex motion in two-dimensional flows. Ann. Rev. Fluid Mech. 15, 345–389 (1983)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bertozzi A.L., Constantin P.: Global regularity for vortex patches. Commun. Math. Phys. 152(1), 19–28 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Burbea J.: Motions of vortex patches. Lett. Math. Phys. 6, 1–16 (1982)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Burton G.R.: Steady symmetric vortex pairs and rearrangements. Proc. R. Soc. Edinb. Sect. A. 108, 269–290 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Burton G.R., Nussenzveig Lopes H.J., Lopes Filho M.C.: Nonlinear Stability for Steady Vortex Pairs. Commun. Math. Phys. 324, 445–463 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    A. Castro, D. Córdoba, J. Gómez-Serrano, Uniformly rotating analytic global patch solutions for active scalars. Ann. PDE 2(1) (2016) (Art. 1)Google Scholar
  7. 7.
    Castro A., Córdoba D., Gómez-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
  8. 8.
    Chae D., Constantin P., Córdoba D., Gancedo F., Wu J.: Generalized surface quasi-geostrophic equations with singular velocities. Commun. Pure Appl. Math. 65(8), 1037–1066 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chemin, J.-Y.: Fluides Parfaits Incompressibles, Astérisque 230 (1995) (Perfect Incompressible Fluids translated by I. Gallagher and D. Iftimie, Oxford Lecture Series in Mathematics and Its Applications, Vol. 14, Clarendon Press-Oxford University Press, New York (1998))Google Scholar
  10. 10.
    Constantin P., Majda A.J., Tabak E.: Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar. Nonlinearity 7(6), 1495–1533 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Córdoba D., Fontelos M.A., Mancho A.M., Rodrigo J.L.: Evidence of singularities for a family of contour dynamics equations. Proc. Natl. Acad. Sci. USA 102(17), 5949–5952 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Deem G.S., Zabusky N.J.: Vortex waves: stationary “V-states”, interactions, recurrence, and breaking. Phys. Rev. Lett. 40(13), 859–862 (1978)ADSCrossRefGoogle Scholar
  13. 13.
    de la Hoz F., Hassainia Z., Hmidi T.: Doubly connected V-states for the generalized surface quasi-geostrophic equations. Arch. Ration. Mech. Anal. 220(3), 1209–1281 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    de la Hoz F., Hmidi T., Mateu J., Verdera J.: Doubly connected V-states for the planar Euler equations. SIAM J. Math. Anal. 48(3), 1892–1928 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Denisov S.A.: The centrally symmetric V-states for active scalar equations. Two-dimensional Euler with cut-off. Commun. Math. Phys. 337, 955–1009 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dritschel D.G.: A general theory for two-dimensional vortex interactions. J. Fluid Mech. 293, 269–303 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gallay Th.: Interaction of vortices in weakly viscous planar flows. Arch. Ration. Mech. Anal. 200, 445–490 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gallay, Th.: Interacting vortex pairs in inviscid and viscous planar flows. In: J. Robinson, J. Rodrigo, et W. Sadowski (eds.) Mathematical Aspects of Fluid Mechanics. London Math. Soc. Lecture Notes Series, vol. 402. Cambridge University Press, Cambridge (2012)Google Scholar
  19. 19.
    Gancedo F.: Existence for the \({\alpha}\)-patch model and the QG sharp front in Sobolev spaces. Adv. Math. 217(6), 2569–2598 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hassainia Z., Hmidi T.: On the V-States for the generalized quasi-geostrophic equations. Commun. Math. Phys. 337(1), 321–377 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Held I., Pierrehumbert R., Garner S., Swanson K.: Surface quasi-geostrophic dynamics. J. Fluid Mech. 282, 1–20 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hmidi T., Mateu J., Verdera J.: Boundary regularity of rotating vortex patches. Arch. Ration. Mech. Anal. 209(1), 171–208 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hmidi, T.,Mateu, J.: Bifurcation of rotating patches from Kirchhoff vortices. Discret. Contin. Dyn. Syst. 36(10), 5401–5422 (2016) Google Scholar
  24. 24.
    Hmidi, T., Mateu, J.: Degenerate bifurcation of the rotating patches. Adv. Math. 302, 799–850 (2016) Google Scholar
  25. 25.
    Juckes, M.: Quasigeostrophic dynamics of the tropopause. J. Armos. Sci. (1994) 2756–2768Google Scholar
  26. 26.
    Kamm, J.R.: Shape and stability of two-dimensional uniform vorticity regions. PhD thesis, California Institute of Technology (1987)Google Scholar
  27. 27.
    Keady G.: Asymptotic estimates for symmetric vortex streets. J. Aust. Math. Soc. Ser. B 26, 487–502 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kirchhoff G.: Vorlesungen uber mathematische Physik. Teubner, Leipzig (1874)zbMATHGoogle Scholar
  29. 29.
    Lamb H.: Hydrodynamics. Dover Publications, New York (1945)zbMATHGoogle Scholar
  30. 30.
    Lapeyre G., Klein P.: Dynamics of the upper oceanic layers in terms of surface quasigeostrophic theory. J. Phys. Oceanogr. 36, 165–176 (2006)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    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)CrossRefzbMATHGoogle Scholar
  32. 32.
    Marchioro C., Pulvirenti M.: Euler evolution for singular initial data and vortex theory. Commun. Math. Phys. 91, 563–572 (1983)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Mateu J., Orobitg J., Verdera J.: Extra cancellation of even Calderón-Zygmund operators and quasiconformal mappings. J. Math. Pures Appl. 9(1), 402–431 (2009)CrossRefzbMATHGoogle Scholar
  34. 34.
    Newton P.K.: The N-Vortex Problem, Analytical Techniques. Springer, New York (2001)CrossRefzbMATHGoogle Scholar
  35. 35.
    Norbury J.: Steady planar vortex pairs in an ideal fluid. Commun. Pure Appl. Math. 28, 679–700 (1975)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Pierrehumbert R.T.: A family of steady, translating vortex pairs with distributed vorticity. J. Fluid Mech. 99, 129–144 (1980)ADSCrossRefzbMATHGoogle Scholar
  37. 37.
    Rodrigo J.L.: On the evolution of sharp fronts for the quasi-geostrophic equation. Commun. Pure Appl. Math. 58(6), 821–866 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Saffman P.G.: Vortex Dynamics. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, New York (1992)Google Scholar
  39. 39.
    Saffman P.G., Szeto R.: Equilibrium shapes of a pair of equal uniform vortices. Phys. Fluids 23(12), 2339–2342 (1980)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    Smets D., Schaftingen J.V.: Desingularization of Vortices for the Euler Equation. Arch. Ration. Mech. Anal. 198, 869–925 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Turkington B.: Corotating steady vortex flows with n-fold symmety. Nonlinear Anal. Theory Methods Appl. 9(4), 351–369 (1985)CrossRefzbMATHGoogle Scholar
  42. 42.
    Wu H.M., Overman E.A. II, Zabusky N.J.: Steady-state solutions of the Euler equations in two dimensions: rotating and translating V-states with limiting cases I. Algorithms ans results. J. Comput. Phys. 53, 42–71 (1984)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Yudovich, V.I.: Non-stationnary flows of an ideal incompressible fluid. Zhurnal Vych Matematika 3, 1032–106 (1963)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.IRMARUniversité de Rennes 1Rennes cedexFrance
  2. 2.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterra, BarcelonaSpain

Personalised recommendations