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Finite amplitude convection between stress-free boundaries; Ginzburg- Landau equations and modulation theory. (English) Zbl 0819.76029
Summary: The stability theory for rolls in stress-free convection at finite Prandtl number is affected by coupling with low wavenumber two- dimensional mean-flow modes. In this work, a set of modified Ginzburg- Landau equations describing the onset of convection is derived which accounts for these additional modes. These equations can be used to extend the modulation equations of A. Zippelius and E. D. Siggia [Phys. Fluids 26, 2905-2915 (1983; Zbl 0537.76016)] describing the breakup of rolls, bringing their stability theory into agreement with the results of F. H. Busse and E. W. Bolton [J. Fluid Mech. 146, 115-125 (1984; Zbl 0561.76052)].

MSC:
76E15 Absolute and convective instability and stability in hydrodynamic stability
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[1] Cross, Physica 10D pp 299– (1984)
[2] Guckenheimer, Nonlinear Oscillations, Dynamical Systems, and bifurcations of Vector Fields (1983) · Zbl 0515.34001 · doi:10.1007/978-1-4612-1140-2
[3] DOI: 10.1017/S0022112083002670 · Zbl 0534.76088 · doi:10.1017/S0022112083002670
[4] DOI: 10.1017/S0022112084001786 · Zbl 0561.76052 · doi:10.1017/S0022112084001786
[5] Busse, Contemp. Math. 56 pp 1– (1986) · doi:10.1090/conm/056/01
[6] DOI: 10.1088/0034-4885/41/12/003 · doi:10.1088/0034-4885/41/12/003
[7] Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations (1983) · Zbl 0507.34003 · doi:10.1007/978-1-4684-0147-9
[8] Arneodo, Turbulence and Chaotic Phenomena in Fluids pp 215– (1984)
[9] DOI: 10.1063/1.864055 · Zbl 0537.76016 · doi:10.1063/1.864055
[10] Zippelius, Phys. Rev. A26 pp 1788– (1982) · doi:10.1103/PhysRevA.26.1788
[11] Spiegel, Woods Hole Oceanog. Inst. Tech. Rept. WHOI-85?36 pp 1– (1985)
[12] Spiegel, Chaos in Astrophysics pp 93– (1985)
[13] DOI: 10.1017/S0022112069000127 · Zbl 0179.57501 · doi:10.1017/S0022112069000127
[14] DOI: 10.1017/S0022112065001271 · Zbl 0134.21801 · doi:10.1017/S0022112065001271
[15] DOI: 10.1103/RevModPhys.49.581 · doi:10.1103/RevModPhys.49.581
[16] DOI: 10.1017/S0022112090003238 · Zbl 0706.76096 · doi:10.1017/S0022112090003238
[17] DOI: 10.1017/S0022112069000176 · Zbl 0187.25102 · doi:10.1017/S0022112069000176
[18] DOI: 10.1063/1.863198 · Zbl 0444.76066 · doi:10.1063/1.863198
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