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Nonlinear evolution of the centrifugal instability using a semilinear model

Published online by Cambridge University Press:  19 June 2020

Eunok Yim Open the ORCID record for Eunok Yim [Opens in a new window] ,
P. Billant Open the ORCID record for P. Billant [Opens in a new window]  and
F. Gallaire Open the ORCID record for F. Gallaire [Opens in a new window]
Eunok Yim*
Affiliation:
LFMI, École Polytechnique Fédérale de Lausanne, 1015Lausanne, Switzerland
P. Billant
Affiliation:
LadHyX, CNRS, École Polytechnique, F-91128Palaiseau CEDEX, France
F. Gallaire
Affiliation:
LFMI, École Polytechnique Fédérale de Lausanne, 1015Lausanne, Switzerland
*
†Email address for correspondence: [email protected]
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Abstract

We study the nonlinear evolution of the axisymmetric centrifugal instability developing on a columnar anticyclone with a Gaussian angular velocity using a semilinear approach. The model consists of two coupled equations: one for the linear evolution of the most unstable perturbation on the axially averaged mean flow and another for the evolution of the mean flow under the effect of the axially averaged Reynolds stresses due to the perturbation. Such a model is similar to the self-consistent model of Mantič-Lugo et al. (Phys. Rev. Lett, vol. 113, 2014, 084501) except that the time averaging is replaced by a spatial averaging. The nonlinear evolutions of the mean flow and the perturbations predicted by this semilinear model are in very good agreement with direct numerical simulations for the Rossby number $Ro=-4$ and both values of the Reynolds numbers investigated: $Re=800$ and $2000$ (based on the initial maximum angular velocity and radius of the vortex). An improved model, taking into account the second-harmonic perturbations, is also considered. The results show that the angular momentum of the mean flow is homogenized towards a centrifugally stable profile via the action of the Reynolds stresses of the fluctuations. The final velocity profile predicted by Kloosterziel et al. (J. Fluid Mech., vol. 583, 2007, pp. 379–412) in the inviscid limit is extended to finite high Reynolds numbers. It is in good agreement with the numerical simulations.

JFM classification

Instability: Nonlinear instability Geophysical and Geological Flows: Rotating flows Vortex Flows: Vortex instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 897 , 25 August 2020 , A34
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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