### Abstract

Rotating Boussinesq convection in a plane layer is governed by two dimensionless groups in addition to the Rayleigh number R: the Prandtl number σ and the Taylor number Ta. Scaled equations for fully nonlinear rotating convection in the limit of rapid rotation and small Prandtl number, where the onset of convection is oscillatory, are derived by considering distinguished limits where σ^{n} Ta^{1}/2 ∼ 1 but σ → 0 and Ta → ∞, for different n > 1. In the resulting asymptotic expansion in powers of Ta ^{-1}/2 the leading-order equations, which are independent of n, are solved to provide an analytic description of fully nonlinear convection. Three distinct asymptotic regimes are identified, distinguished by the relative importance of the subdominant buoyancy and inertial terms. For the most interesting case, n = 4, the stability of different planforms near onset is investigated using a double expansion in powers of Ta ^{-1}/8 and the amplitude of convection ε. The lack of a buoyancy term at leading order demands that the perturbation expansion be continued through six orders to derive amplitude equations determining the dynamics. The case n = 1 is also analysed. The relevance of this theory to experimental results is briefly discussed.

Original language | English |
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Pages (from-to) | 61-80 |

Number of pages | 20 |

Journal | Journal of Fluid Mechanics |

Volume | 428 |

DOIs | |

Publication status | Published - 10 Feb 2001 |

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### ASJC Scopus subject areas

- Condensed Matter Physics
- Mechanics of Materials
- Mechanical Engineering

### Cite this

**Rapidly rotating thermal convection at low prandtl number.** / Dawes, J. H.P.

Research output: Contribution to journal › Article

*Journal of Fluid Mechanics*, vol. 428, pp. 61-80. https://doi.org/10.1017/S002211200000255X

}

TY - JOUR

T1 - Rapidly rotating thermal convection at low prandtl number

AU - Dawes, J. H.P.

PY - 2001/2/10

Y1 - 2001/2/10

N2 - Rotating Boussinesq convection in a plane layer is governed by two dimensionless groups in addition to the Rayleigh number R: the Prandtl number σ and the Taylor number Ta. Scaled equations for fully nonlinear rotating convection in the limit of rapid rotation and small Prandtl number, where the onset of convection is oscillatory, are derived by considering distinguished limits where σn Ta1/2 ∼ 1 but σ → 0 and Ta → ∞, for different n > 1. In the resulting asymptotic expansion in powers of Ta -1/2 the leading-order equations, which are independent of n, are solved to provide an analytic description of fully nonlinear convection. Three distinct asymptotic regimes are identified, distinguished by the relative importance of the subdominant buoyancy and inertial terms. For the most interesting case, n = 4, the stability of different planforms near onset is investigated using a double expansion in powers of Ta -1/8 and the amplitude of convection ε. The lack of a buoyancy term at leading order demands that the perturbation expansion be continued through six orders to derive amplitude equations determining the dynamics. The case n = 1 is also analysed. The relevance of this theory to experimental results is briefly discussed.

AB - Rotating Boussinesq convection in a plane layer is governed by two dimensionless groups in addition to the Rayleigh number R: the Prandtl number σ and the Taylor number Ta. Scaled equations for fully nonlinear rotating convection in the limit of rapid rotation and small Prandtl number, where the onset of convection is oscillatory, are derived by considering distinguished limits where σn Ta1/2 ∼ 1 but σ → 0 and Ta → ∞, for different n > 1. In the resulting asymptotic expansion in powers of Ta -1/2 the leading-order equations, which are independent of n, are solved to provide an analytic description of fully nonlinear convection. Three distinct asymptotic regimes are identified, distinguished by the relative importance of the subdominant buoyancy and inertial terms. For the most interesting case, n = 4, the stability of different planforms near onset is investigated using a double expansion in powers of Ta -1/8 and the amplitude of convection ε. The lack of a buoyancy term at leading order demands that the perturbation expansion be continued through six orders to derive amplitude equations determining the dynamics. The case n = 1 is also analysed. The relevance of this theory to experimental results is briefly discussed.

UR - http://www.scopus.com/inward/record.url?scp=0035069968&partnerID=8YFLogxK

U2 - 10.1017/S002211200000255X

DO - 10.1017/S002211200000255X

M3 - Article

VL - 428

SP - 61

EP - 80

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

ER -