Abstract
The close analogy between instabilities in nonlinear systems driven far from equilibrium, and equilibrium phase transitions in condensed-matter physics, is now well documented experimentally as well as theoretically. This idea was fathered by Landau [1] and developped by several authors in the context of hydrodynamics [2], electric circuits, semiconductors, nonlinear optics and chemical instabilities [3].
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
L. Landau, “On the theory of phase transitions”, “On the problem of turbulence”, Collected Papers of L. Landau, Gordon and Breach (1967).
For a review, see P.H. Coullet and E.A. Spiegel, SIAM J. Appl. Math. 43, 775–821 (1983).
For a review, see A. Nitzan, P. Ortoleva, J. Deutch and J. Ross, J. Chem. Phys. 61, 1056–1074 (1974).
For a review, see J.P. Gollub and H.L. Swinney, Hydrodynamic Stability and the Transition to Turbulence, Topics in applied physics 45, Springer Verlag (1981)
J.E. Wesfreid and S. Zaleski, Cellular Structures in Instabilities. Lecture notes in physics, Springer Verlag (1984).
A.C. Newell, Lectures in Applied Math. 15, 157–163 (1974).
G.B. Witham, Linear and Nonlinear Waves, Wiley (1974).
D.J. Benney and A.C. Newell, J. Math. Phys. 46, 133–139 (1967).
P. Ortoleva and J. Ross, J. Chem. Phys. 58, 5673–5680 (1973).
L.N. Howard and N. Kopell, Stud. Appl. Math. 56, 95–145 (1977).
Y. Kuramoto and T. Tsuzuki, Prog. Theor. Phys. 55, 356–369 (1976).
Y. Pomeau and P. Manneville, J. Physique Lettres 40, 609–612(1979)
M.C. Cross, Phys. Rev. A27, 490–498 (1983).
M.C. Cross and A.C. Newell, Physica 10D, 229–328 (1984).
P. Tabeling, J. Physique Lettres 44, 665–672 (1983).
H. Brand and M.C. Cross, Phys. Rev. A27, 1237–1239 (1983).
H. Brand, Prog. Theor. Phys. 71, 1096–1099 (1984).
P.W. Anderson, Basic Notions of Codensed Matter Physics, Frontiers in Physics, Benjamin (1984).
Y. Kuramoto, Prog. Theor. Phys. 71, 1182–1196 (1984).
S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Clarendon Press (1961).
J. Swift and P.C. Hohenberg, Phys. Rev. A15, 319–328 (1979).
A.C. Newell and J.A. Whitehead, J. Fluid Mech. 38, 279–303 (1969).
L.A. Segel, J. Fluid Mech. 38, 203–224 (1969).
F. Busse, J. Fluid Mech. 52, 97–112 (1972).
E.D. Siggia and A. Zippelius, Phys. Rev. Lett. 47, 835–838 (1981).
R.M. Clever and F.H. Busse, J. Fluid. Mech. 65, 625–645 (1974).
B.F. Edwards and A.L. Fetter, Phys. Fluids 27, 2795–2802 (1984).
S. Fauve, Instabilités convectives et équations d’amplitude, Ecole de Goutelas, Observatoire de Meudon (1983).
P. Coullet and S. Fauve, Phys. Rev. Lett. 55, 2857–2859 (1985).
A. Schlüter, D. Lortz, and F. Busse, J. Fluid Mech. 23, 129–144 (1965).
V.I. Arnold, Supplementary Chapters to the Theory of Ordinary differential Equations, Nauka (1978).
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical System and Bifurcation of Vector Fields, Springer Verlag (1984).
Note that in this model, the large scale horizontal velocity field should be of the form Q(z) ψ;(Y,T), where Q(z) satisfies the boundary conditions.
E.W. Bolton, F.H. Busse and R.M. Clever, Preprint (1985).
F.B. Lipps, J. Fluid Mech. 75, 113–148 (1976).
J.B. McLaughlin and S.A. Orszag, J. Fluid. Mech. 122 (1982).
G.E. Willis and J.W. Deardorff, J. Fluid Mech. 44, 661–672 (1970).
A.R. Bishop, J.A. Krumhansl and S.E. Trullinger, Physica 1D, 1–44 (1980).
G.B. Stokes, Trans. Camb. Phil. Soc. 8441, 197 (1847).
T.B. Benjamin and J.E. Feir, J. Fluid Mech. 27, 417–430 (1967).
J.T. Stuart and R.C. Di Prima, Proc. Roy. Soc. A362, 27–41 (1978).
R.C. Di Prima and R.N. Grannick, IUTAM Symp. on Instabilities of Continuous Systems, H. Leipholtz ed. Springer Verlag (1971).
C.S. Bretherton and E.A. Spiegel, Phys. Lett. 96A, 152–156 (1983).
Y. Demay and G. Iooss, J. Mécanique (numéro spécial 1984).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1987 D. Reidel Publishing Company
About this chapter
Cite this chapter
Fauve, S. (1987). Large Scale Instabilities of Cellular Flows. In: Tirapegui, E., Villarroel, D. (eds) Instabilities and Nonequilibrium Structures. Mathematics and Its Applications, vol 33. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-3783-3_4
Download citation
DOI: https://doi.org/10.1007/978-94-009-3783-3_4
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-8183-2
Online ISBN: 978-94-009-3783-3
eBook Packages: Springer Book Archive