Spatially-periodic steady solutions to the three-dimensional Navier-Stokes equation with the ABC-force

Physica D: Nonlinear Phenomena

Volumn 128, Issue 2-4, 1999, Pages 250-272

  Podvigina, Olga M ^a,b  

^a UNIVERSITY OF SYDNEY   (Australia)

^b INSTITUTE OF EARTHQUAKE PREDICTION THEORY AND MATHEMATICAL GEOPHYSICS   (Russian Federation)

Author keywords

Bifurcation; Navier Stokes equation; Steady solution; Symmetry

Indexed keywords

EID: 0347596746     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-2789(98)00292-9     Document Type: Article

References (33)

1. V.I. Arnold, Sur la topologie des écoulements stationnaires des fluides parfaits, Comptes Rendus Acad. Sci. Paris 261 (1965) 17-20 (in French).
2. T. Dombre, U. Frisch, J.M. Greene, M. Hénon, A. Mehr, A. Soward, Chaotic streamlines in the ABC flows, J. Fluid Mech. 167 (1986) 353-391.
3. M. Hénon, Sur la topologie des lignes de courant dans un cas particulier, Comptes Rendus Acad. Sci. Paris 262 (1966) 312-314 (in French).
4. B. Galanti, P.L. Sulem, A. Pouquet, Linear and Non-Linear Dynamos Associated With ABC Flows, Geophys. Astrophys. Fluid Dynamics 66 (1992) 183-208.
5. M.M. Vishik, On magnetic field generation by a three-dimensional steady flow of conducting fluid at high magnetic Reynolds number, Izvestia Acad. Sci. USSR, Fizika Zemli 3 (1988) 3-12 (in Russian).
6. M.M. Vishik, Magnetic field generation by the motion of a highly conducting fluid, Geophys. Astrophys. Fluid Dynamics 48 (1989) 151-167.
7. I. Klapper, L.S. Young, Rigorous bounds on the fast dynamo growth rate involving topological entropy, Comm. Math. Phys. 173 (1995) 623-646.
8. S. Childress, New Solutions of the Kinematic Dynamo Problem, J. Math. Phys. 11 (1970) 3063.
9. V.I. Arnold, E.I. Korkina, The growth of a magnetic field in a three-dimensional incompressible flow, Vestnik Moscow State Univ., Ser. Math. 3 (1983) 43-46 (in Russian).
10. V.I. Arnold, On the evolution of magnetic field under the action of advection and diffusion, in: V.M. Tikhomirov (Ed.), Some Problems of Modern Analysis, Moscow Univ. Press (1984), 8-21 (in Russian).
11. S. Childress, A.D. Gilbert, Stretch, twist, fold: the fast dynamo, Springer, Berlin, 1995.
12. D.J. Galloway, U. Frisch, A numerical investigation of magnetic field generation in a flow with chaotic streamlines, Geophys. Astrophys. Fluid Dynamics 29 (1984) 13-18.
13. D.J. Galloway, U. Frisch, Dynamo action in a family of flows with chaotic streamlines, Geophys. Astrophys. Fluid Dynamics 36 (1986) 53-83.
14. B. Galanti, P.L. Sulem, A. Pouquet, Influence of the Period of an ABC Flow on its Dynamo Action, in: M.R.E. Proctor, P.C. Matthews, A.M. Rucklidge (Eds.), Solar and Planetary Dynamos, Proc. of a NATO Advanced Study Institute held at the Isaac Newton Institute, Cambridge, September 1992, Cambridge University Press, 1993, pp. 99-103.
15. A.D. Gilbert, Magnetic field evolution in steady chaotic flows, Phil. Trans. Roy. Soc. London 339 (1992) 627-656.
16. D.J. Galloway, U. Frisch, A note on the stability of a family of space-periodic Beltrami flows, J. Fluid Mech. 180 (1987) 557-564.
17. O. Podvigina, A. Pouquet, On the non-linear stability of the 1:1:1 ABC flow, Physica D 75 (1994) 471-508.
18. O. Zheligovsky (Podvigina), A. Pouquet, Hydrodynamic Stability of the ABC Flow, in: M.R.E. Proctor, P.C. Matthews, A.M. Rucklidge (Eds.), Solar and Planetary Dynamos, Proceedings of a NATO Advanced Study Institute held at the Isaac newton Institute, Cambridge, September 1992, Cambridge University Press, 1993, pp. 347-354.
19. D. Gottlieb, S.A. Orszag, Numerical Analyssi of Spectral Methods, SIAM CBMS-NSF Regional Conference Seires in Applied Math, 1997.
20. C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods in Fluid Dynamics, Springer, Berlin, 1988.
21. J.P. Boyd, Chebyshev & Fourier Spectral Methods, Springer, Berlin, 1989.
22. S. Friedlander, M.M. Vishik, Instability criteria for the Flow of an Inviscid Incompressible Fluid, Phys. Rev. Lett. 66 (1991) 2204-2206.
23. M.M. Vishik, S. Friedlander, Dynamo Theory Methods for Hydrodynamic Stability, J. Maths. Pures Appl. 72 (1993) 145-180.
24. O.M. Podvigina, V.A. Zheligovsky, An optimized iterative method for numerical solution of large systems of equations based on the extremal property of zeroes of Chebyshev polynomials, J. Scientific Computing 12(4) (1997) 433-464.
25. A.N. Kolmogorov, S.V. Fomin, Introductory real analysis, Dover Publications, NY, 1975.
26. A.Yu. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, NY, 1995.
27. H.S.M. Coxeter, Regular Polytopes, Methuen and Co., London, 1948.
28. M. Golubitsky, I. Stewart, D.G. Schaeffer, Singularities and groups in Bifurcation Theory, vol. 2, Applied Mathematical Sciences, 69, Springer, 1988.
29. M. Hall, Theory of Groups, The Macmillan Company, NY, 1959.
30. M.V. Jaric, L. Michel, R.T. Sharp, Zeros of covariant vector fields for the point groups: invariant formulation, J. Physique 45 (1984) 1-47.
31. O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon & Breach, NY, 1969.
32. M.E. Taylor, Pseudodifferential Operators, Princeton University Press, 1981.
33. R. Temam, Navier-Stokes equations, Studies in Mathematics and its Applications 2, Elsevier, NY, 1984.