Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com


An Approach to the Modeling of Nonlinear Structures in Systems with a Multi-component Convection

Discontinuity, Nonlinearity, and Complexity 4(3) (2016) 323--331 | DOI:10.5890/DNC.2016.09.008

Sergey Kozitskiy

Department of Oceanic and Atmospheric Physics, Il’ichev Pacific Oceanological Institute, 43 Baltiyskay str.Vladivostok, 690041, Russia

Download Full Text PDF

 

Abstract

We consider 3D multi-component convection in a horizontally infinite layer of an uncompressible fluid slowly rotating around a vertical axis. A family of CGLE type amplitude equations is derived by multiple-scaled method in the neighborhood of Hopf bifurcation points. We numerically simulate a case of the three-mode convection at large Rayleigh numbers. It was shown that the convection typically takes a form of hexagonal structures for a localized initial conditions. The rotation of the system prevents the spread of the convective structures on the entire area. The approach to the modeling of the Saturn’s polar hexagon on the basis of amplitude equations is discussed.

Acknowledgments

This work is supported by RFBR grant 14-05-00017.

References

  1. [1]  Newell, A.C. and Whitehead, J.A. (1968), Finite bandwidth, finite amplitude convection. Journal of Fluid Mechanics, 38, 279-303.
  2. [2]  Bretherton, C.S. and Spiegel, E.A. (1983), Intermittency through modulational instability. Physics Letters, 96A, 152- 156.
  3. [3]  Kozitskiy, S.B. (2000), Amplitude equations for a system with thermohaline convection, Journal of Applied Mechanics and Technical Physics 41(3), 429-438.
  4. [4]  Balmforth, N.J. and Biello, J.A. (1998), Double diffusive instability in a tall thin slot, Journal of Fluid Mechanics 375, 203-233.
  5. [5]  Kozitskiy, S.B. (2012),Model of three dimensional double-diffusive convection with cells of an arbitrary shape, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 4, 43-61.
  6. [6]  Kozitskiy, S.B. (2014), Structures in 3D double-diffusive convection and possible approach to the Saturn's polar hexagon modeling, E-print arXiv:1405.3020 [nlin.PS] (http://arxiv.org/abs/1405.3020).
  7. [7]  Weiss, N.O. (1981), Convection in an imposed magnetic field. part 1. the development of nonlinear convection, Journal of Fluid Mechanics, 108, 247-272.
  8. [8]  Nayfeh, A.H. (1993), Introduction to perturbation techniques, John Wiley & Sons, New York-Chichester-Brisbane- Toronto.
  9. [9]  Kozitskiy, S.B. (2005), Fine structure generation in double-diffusive system, Physical Review E, 72(5), 056309-1- 056309-6.
  10. [10]  Kozitskiy, S.B. (2010), Amplitude equations for three-dimensional roll-type double-diffusive convection with an arbitrary cell width in the neighborhood of hopf bifurcation points, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 4, 13-24.
  11. [11]  Cox, S.M. and Matthews, P.C. (2002), Exponential time differencing for stiff systems, Journal of Computational Physics, 176, 430-455.
  12. [12]  Leconte, J. and Chabrier, G. (2012), A new vision of giant planet interiors: Impact of double diffusive convection, Astron. Astrophys, 540, A20.