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Dynamics of vortex line in presence of stationary vortex

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Abstract

The motion of a thin vortex with infinitesimally small vorticity in the velocity field created by a steady straight vortex is studied. The motion is governed by non-integrable PDE generalizing the Nonlinear Schrodinger equation (NLSE). Situation is essentially different in a co-rotating case, which is analog of the defocusing NLSE and a counter-rotating case, which can be compared with the focusing NLSE. The governing equation has special solutions shaped as rotating helixes. In the counter-rotating case all helixes are unstable, while in the co-rotating case they could be both stable and unstable. Growth of instability of counter-rotating helix ends up with formation of singularity and merging of vortices. The process of merging goes in a self-similar regime. The basic equation has a rich family of solitonic solutions. Analytic calculations are supported by numerical experiment.

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References

  1. Crow S.C.: Stability theory for a pair of trailing vortices. AIAA J. 8, 2172–2179 (1970)

    Article  Google Scholar 

  2. Zakharov V.E.: Wave collapse. Phys.-Uspekhi 155, 529–533 (1988)

    Google Scholar 

  3. Zakharov V.E.: Quasi-two-dimensional hydrodynamics and interaction of vortex tubes. In: Passot, T., Sulem, P.-L. (eds) Lecture Notes in Physics, Vol. 536, pp. 369. Springer, Berlin (1999)

    Google Scholar 

  4. Klein R., Maida A., Damodaran K.: Simplified analysis of nearly parallel vortex filaments. J. Fluid Mech. 288, 201–248 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  5. Lions P.L., Maida A.J.: Equilibrium statistical theory for nealy parallel vortex filaments. Comm. Pure Appl. Math. 53(1), 76–142 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  6. Maida A.J., Bertozzi A.L.: Vorticity and incompressible flow. Cambridge University Press, Cambridge, MA (2002)

    Google Scholar 

  7. Kerr R.M.: Evidence for a singularity of the three-dimensional incompressible Euler equation. Phys. Fluids A Fluid Dyn. 5, 1725 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  8. How T.Y., Li R.: Blowup or no blowup? The interplay between theory and numerics. Physica D 237, 1937–1944 (2008)

    Article  MathSciNet  Google Scholar 

  9. Hashimoto H.: A solution on vortex filaments. Fluid Dyn. Res. 3, 1–12 (1972)

    Article  Google Scholar 

  10. Zakharov V.E., Takhtajan L.A.: Equivalence of a nonlinear Schrodinger equation and a Geizenberg ferromagnet equation. Theor. Math. Phys. 38(1), 26–35 (1979)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Arizona, Tucson, AZ, USA

    Vladimir E. Zakharov

  2. P.N. Lebedev Physical Institute RAS, Moscow, Russia

    Vladimir E. Zakharov

Authors
  1. Vladimir E. Zakharov
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Correspondence to Vladimir E. Zakharov.

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Communicated by H. Aref

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Zakharov, V.E. Dynamics of vortex line in presence of stationary vortex. Theor. Comput. Fluid Dyn. 24, 377–382 (2010). https://doi.org/10.1007/s00162-009-0164-z

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  • DOI: https://doi.org/10.1007/s00162-009-0164-z

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