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144<br />
Three-dimensional stability of vortex arrays in a stratified<br />
fluid<br />
Axel Deloncle ∗ , Paul Billant and Jean-Marc Chomaz<br />
The three-dimensional linear stability of classical vortex configurations (Von Karman<br />
street and double symmetric row) is investigated through a theoretical and numerical<br />
analysis in the case of a strongly stratified fluid.<br />
For strong stratification, recent studies have shown that columnar vertical vortex<br />
pairs are subject to a 3D zigzag instability 1,2 which consists in a bending of the<br />
vortices (figure 1–left) and leads to the emergence of horizontal layers.<br />
We demonstrate that both the Von Karman street and a double symmetric row of<br />
columnar vertical vortices are also unstable to the zigzag instability. By means of an<br />
asymptotic theory in the limit of long-vertical wavelength and well-separated vortices,<br />
the most unstable wavelength is found to be proportional to bFh, where b is the<br />
separation distance between the vortices and Fh the horizontal Froude number (Fh =<br />
Γ/πa 2 N with Γ the circulation of the vortices, a their core radius and N the Brunt-<br />
Väisälä frequency). The maximum growthrate is independent of the stratification<br />
and only proportional to the strain S =Γ/2πb 2 . A numerical linear stability analysis<br />
fully confirms the theoretical predictions (figure 1–right).<br />
The non-linear evolution of these vortices arrays as well as other random configurations<br />
shows that the zigzag instability ultimately slices the flow into horizontal<br />
layers. These results demonstrate that the zigzag instability is a generic instability.<br />
It may explain the observations of layered structures in experiments 3 and numerical<br />
simulations 4 of stratified turbulence.<br />
∗ LadHyX, Ecole Polytechnique, CNRS, 91128 Palaiseau, France.<br />
1 Billant and Chomaz, J. Fluid Mech. 418, 167 (2000).<br />
2 Otheguy et al., J. Fluid Mech. in press.<br />
3 Praud et al., J. Fluid Mech. 522, 1 (2005).<br />
4 Waite and Bartello, J. Fluid Mech. 517, 281 (2004).<br />
2πb 2 σ/Γ<br />
5<br />
4<br />
3<br />
2<br />
1<br />
0<br />
5 10 15 20 25<br />
k bF<br />
z h<br />
Figure 1: Von Karman street. (left) sketch of the zigzag instability. (right) growthrate<br />
2πb 2 σ/Γ of the zigzag instability versus the vertical wavenumber kzbFh. ---: asymptotic<br />
theory; Numerics with a 256x640 resolution: Fh =0.2 △, Fh =0.4 ▽
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