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Nonlinear evolution of the zigzag instability in stratified fluids: a shortcut on the route to dissipation

Published online by Cambridge University Press:  06 March 2008

AXEL DELONCLE ,
PAUL BILLANT  and
JEAN-MARC CHOMAZ
AXEL DELONCLE
Affiliation:
LadHyX, CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex, Franceaxel.deloncle@ladhyx.polytechnique.fr
PAUL BILLANT
Affiliation:
LadHyX, CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex, Franceaxel.deloncle@ladhyx.polytechnique.fr
JEAN-MARC CHOMAZ
Affiliation:
LadHyX, CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex, Franceaxel.deloncle@ladhyx.polytechnique.fr
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Abstract

We present high-resolution direct numerical simulations of the nonlinear evolution of a pair of counter-rotating vertical vortices in a stratified fluid for various high Reynolds numbers Re and low Froude numbers Fh. The vortices are bent by the zigzag instability producing high vertical shear. There is no nonlinear saturation so that the exponential growth is stopped only when the viscous dissipation by vertical shear is of the same order as the horizontal transport, i.e. when /Re=O(1) where is the maximum horizontal enstrophy non-dimensionalized by the vortex turnover frequency. The zigzag instability therefore directly transfers the energy from large scales to the small dissipative vertical scales. However, for high Reynolds number, the vertical shear created by the zigzag instability is so intense that the minimum local Richardson number Ri decreases below a threshold of around 1/4 and small-scale Kelvin–Helmholtz instabilities develop. We show that this can only occur when is above a threshold estimated as 340. Movies are available with the online version of the paper.

JFM classification

Vortex Flows: Vortex instability Geophysical and Geological Flows: Stratified flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 599 , 25 March 2008 , pp. 229 - 239
Copyright
Copyright © Cambridge University Press 2008

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References

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Deloncle supplementary movies

Movie 1. Pseudo-spectral numerical simulation of the nonlinear evolution of a pair of counter-rotating vertical vortices in a stratified fluid for a Froude number F_h = 0.66 and a Reynolds number Re = 1060 (run labelled F0.6R1 in the paper). The vortices, initially straight along the vertical, are bent as a whole by the zigzag instability producing high vertical shear. Vertical vorticity isosurfaces are plotted: red and deep blue contours represent respectively plus and minus 60% of the vertical average of the maximum vertical vorticity in each horizontal plane. Transparent yellowish and bluish isosurfaces are the same, but for a 10% level.

Download Deloncle supplementary movies(Video)
Video 1.1 MB

Deloncle supplementary movies

Movie 2. Similar to Movie 1 but for the simulation F0.6R3 (F_h = 0.66, Re = 3180). The small-scale structures that appear correspond to Kelvin-Helmholtz instabilities.

Download Deloncle supplementary movies(Video)
Video 1.8 MB