### Abstract

A theory, which should have widespread application, is developed to treat the statics and slow dynamics of patterns of convective rolls encountered in large aspect ratio Rayleigh-Bénard boxes. For case of presentation the theory is developed using model equations. Wavenumber selection, the shape of patterns, stability and the time dependence resulting from long wavelength instabilities are discussed. The effects of adding the local mean drift important in low Prandtl number situations are investigated. Our theory includes the notion of the Busse stability balloon, reduces near critical values of the stress parameter to the Newell-Whitehead-Segel equations, and contains the Pomeau-Manneville phase equation. It also gives a detailed description of the way in which the field amplitude, which is slaved to the phase gradient away from threshold, becomes an independent order parameter near the critical point.

Original language | English (US) |
---|---|

Pages (from-to) | 299-328 |

Number of pages | 30 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 10 |

Issue number | 3 |

DOIs | |

State | Published - 1984 |

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### ASJC Scopus subject areas

- Applied Mathematics
- Statistical and Nonlinear Physics

### Cite this

*Physica D: Nonlinear Phenomena*,

*10*(3), 299-328. https://doi.org/10.1016/0167-2789(84)90181-7

**Convection patterns in large aspect ratio systems.** / Cross, M. C.; Newell, Alan C.

Research output: Contribution to journal › Article

*Physica D: Nonlinear Phenomena*, vol. 10, no. 3, pp. 299-328. https://doi.org/10.1016/0167-2789(84)90181-7

}

TY - JOUR

T1 - Convection patterns in large aspect ratio systems

AU - Cross, M. C.

AU - Newell, Alan C

PY - 1984

Y1 - 1984

N2 - A theory, which should have widespread application, is developed to treat the statics and slow dynamics of patterns of convective rolls encountered in large aspect ratio Rayleigh-Bénard boxes. For case of presentation the theory is developed using model equations. Wavenumber selection, the shape of patterns, stability and the time dependence resulting from long wavelength instabilities are discussed. The effects of adding the local mean drift important in low Prandtl number situations are investigated. Our theory includes the notion of the Busse stability balloon, reduces near critical values of the stress parameter to the Newell-Whitehead-Segel equations, and contains the Pomeau-Manneville phase equation. It also gives a detailed description of the way in which the field amplitude, which is slaved to the phase gradient away from threshold, becomes an independent order parameter near the critical point.

AB - A theory, which should have widespread application, is developed to treat the statics and slow dynamics of patterns of convective rolls encountered in large aspect ratio Rayleigh-Bénard boxes. For case of presentation the theory is developed using model equations. Wavenumber selection, the shape of patterns, stability and the time dependence resulting from long wavelength instabilities are discussed. The effects of adding the local mean drift important in low Prandtl number situations are investigated. Our theory includes the notion of the Busse stability balloon, reduces near critical values of the stress parameter to the Newell-Whitehead-Segel equations, and contains the Pomeau-Manneville phase equation. It also gives a detailed description of the way in which the field amplitude, which is slaved to the phase gradient away from threshold, becomes an independent order parameter near the critical point.

UR - http://www.scopus.com/inward/record.url?scp=0021390251&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0021390251&partnerID=8YFLogxK

U2 - 10.1016/0167-2789(84)90181-7

DO - 10.1016/0167-2789(84)90181-7

M3 - Article

AN - SCOPUS:0021390251

VL - 10

SP - 299

EP - 328

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 3

ER -