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Reynolds stresses and mean fields generated by pure waves: applications to shear flows and convection in a rotating shell

Published online by Cambridge University Press:  25 April 2008

E. PLAUT ,
Y. LEBRANCHU ,
R. SIMITEV  and
F. H. BUSSE
E. PLAUT
Affiliation:
LEMTA, Nancy-Université & CNRS, 2 av. Fosrêt de Haye, 54516 Vandoeuvre cedex, France
Y. LEBRANCHU
Affiliation:
LEMTA, Nancy-Université & CNRS, 2 av. Fosrêt de Haye, 54516 Vandoeuvre cedex, France
R. SIMITEV
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow, G12 8QW, UK
F. H. BUSSE
Affiliation:
Institute of Physics, University of Bayreuth, 95440 Bayreuth, Germany
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Abstract

A general reformulation of the Reynolds stresses created by two-dimensional waves breaking a translational or a rotational invariance is described. This reformulation emphasizes the importance of a geometrical factor: the slope of the separatrices of the wave flow. Its physical relevance is illustrated by two model systems: waves destabilizing open shear flows; and thermal Rossby waves in spherical shell convection with rotation. In the case of shear-flow waves, a new expression of the Reynolds–Orr amplification mechanism is obtained, and a good understanding of the form of the mean pressure and velocity fields created by weakly nonlinear waves is gained. In the case of thermal Rossby waves, results of a three-dimensional code using no-slip boundary conditions are presented in the nonlinear regime, and compared with those of a two-dimensional quasi-geostrophic model. A semi-quantitative agreement is obtained on the flow amplitudes, but discrepancies are observed concerning the nonlinear frequency shifts. With the quasi-geostrophic model we also revisit a geometrical formula proposed by Zhang to interpret the form of the zonal flow created by the waves, and explore the very low Ekman-number regime. A change in the nature of the wave bifurcation, from supercritical to subcritical, is found.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 602 , 10 May 2008 , pp. 303 - 326
Copyright
Copyright © Cambridge University Press 2008

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