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Onset of convection in a moderate aspect-ratio rotating cylinder: Eckhaus–Benjamin–Feir instability

Published online by Cambridge University Press:  15 October 2007

J. M. LOPEZ ,
F. MARQUES ,
I. MERCADER  and
O. BATISTE
J. M. LOPEZ
Affiliation:
Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, USA
F. MARQUES
Affiliation:
Departament de Física Aplicada, Univ. Politècnica de Catalunya, Barcelona 08034, Spain
I. MERCADER
Affiliation:
Departament de Física Aplicada, Univ. Politècnica de Catalunya, Barcelona 08034, Spain
O. BATISTE
Affiliation:
Departament de Física Aplicada, Univ. Politècnica de Catalunya, Barcelona 08034, Spain
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Abstract

A numerical study of the onset of thermal convection in a rotating circular cylinder of radius-to-depth ratio equal to four is considered in a regime dominated by the Coriolis force where the onset is to so-called wall modes. The wall modes consist of hot and cold pairs of thermal plumes rising and descending in the cylinder wall boundary layer, forming an essentially one-dimensional pattern characterized by the number of hot/cold plume pairs, m. In the limit of zero centrifugal force, this onset of convection at a critical temperature difference across the depth of the cylinder is via a symmetry-breaking supercritical Hopf bifurcation which leads to retrograde precession of the pattern with respect to the rotation of the cylinder. For temperature differences greater than critical, a number of distinct wall modes, distinguished by m, coexist and are stable. Their dynamics are controlled by an Eckhaus–Benjamin–Feir instability, the most basic features of which had been captured by a complex Ginzburg–Landau equation model. Here, we analyse this instability in rotating convection using direct numerical simulations of the Navier–Stokes equations in the Boussinesq approximation. Several properties of the wall modes are computed, extending the results to far beyond the onset of convection. Extensive favourable comparisons between our numerical results and previous experimental observations and complex Ginzburg–Landau model results are made.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 590 , 10 November 2007 , pp. 187 - 208
Copyright
Copyright © Cambridge University Press 2007

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