Zigzag instability of vortex arrays in a stratified fluid

We investigate the three-dimensional linear stability of classical vortex configurations (Von Karman street, double symmetric row) in a strongly stratified fluid. By means of an asymptotic theory in the limit of long-vertical wavelength and well-separated vortices, we demonstrate that both the Von Karman street and a double symmetric row of columnar vertical vortices are unstable to the zigzag instability. This instability corresponds to a bending of the vortices with almost no internal deformation and ultimately slices the flow into horizontal layers. The most unstable wavelength is found to be proportional to $bF_h$, where $b$ is the separation distance between the vortices and $F_h$ the horizontal Froude number ($Fh=\Gamma/\pi a^2N$ with $\Gamma$ the circulation of the vortices, $a$ their core radius and $N$ the Brunt-V\"ais\"al\"a frequency). The maximum growth rate is independent of the intensity of the stratification and only proportional to the strain $S = \Gamma/2\pi b^2$. These results may explain the formation of layers observed in stratified turbulence.