Mode interactions in large aspect ratio convection.

*(English)*Zbl 0902.76035Summary: Mode interaction between odd and even modes in two-dimensional Boussinesq convection in a box is revisited. It is noted that in the large aspect ratio limit the structure of the amplitude equations depends on the boundary conditions applied at the sidewalls, however distant. With no-slip sidewall boundary conditions the equations approach those for an unbounded layer with periodic boundary conditions; this is not the case for free-slip boundary conditions. Thus only in the former case the large aspect ratio system can be considered as a small perturbation of the unbounded system. The reasons for the different large aspect ratio limit are traced to the presence of “hidden” symmetries in the stress-free case. Homotopic continuation is used to extend these results to other types of boundary conditions.

##### MSC:

76E15 | Absolute and convective instability and stability in hydrodynamic stability |

76E30 | Nonlinear effects in hydrodynamic stability |

80A20 | Heat and mass transfer, heat flow (MSC2010) |

##### Keywords:

two-dimensional Boussinesq convection in box; hidden symmetries; homotopic continuation; no-slip sidewall boundary conditions; stress-free case
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\textit{P. Hirschberg} and \textit{E. Knobloch}, J. Nonlinear Sci. 7, No. 6, 537--556 (1997; Zbl 0902.76035)

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