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Theoretical analysis of the zigzag instability of a vertical co-rotating vortex pair in a strongly stratified fluid

Published online by Cambridge University Press:  25 July 2007

PANTXIKA OTHEGUY ,
PAUL BILLANT  and
JEAN-MARC CHOMAZ
PANTXIKA OTHEGUY
Affiliation:
LadHyX, CNRS, École Polytechnique, F–91128 Palaiseau Cedex, France
PAUL BILLANT
Affiliation:
LadHyX, CNRS, École Polytechnique, F–91128 Palaiseau Cedex, France
JEAN-MARC CHOMAZ
Affiliation:
LadHyX, CNRS, École Polytechnique, F–91128 Palaiseau Cedex, France
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Abstract

A long-wavelength stability analysis of two co-rotating Gaussian vertical vortices in an inviscid strongly stratified fluid is conducted for vortices separated by a large distance b compared to their radius a (b ≫ a). This analysis predicts and explains the zigzag instability found by a numerical stability analysis in a companion paper (Otheguy, Chomaz & Billant, J. Fluid. Mech. vol. 553, 2006, p. 253). The zigzag instability results from the coupling between the bending perturbations of each vortex and the external strain that one vortex induces on the other S = Γ/2 π b2, where Γ is the circulation of the vortices. The analysis predicts that the maximum growth rate of the instability is twice the strain S and that the most unstable vertical wavelength λ scales as the buoyancy length, defined by LB = Γ/πaN, multiplied by the ratio b/a, i.e. λ ∝ Fhb, where Fh = Γ/πa2N is the horizontal Froude number. The asymptotic results are in very good agreement with the numerical results.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 584 , 10 August 2007 , pp. 103 - 123
Copyright
Copyright © Cambridge University Press 2007

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