Elliptic and zigzag instabilities on co-rotating vertical vortices in a stratified fluid

Abstract

We present a three-dimensional linear stability analysis of a couple of co-rotating vertical vortices in a stratified fluid. When the fluid is non-stratified, the two vortices are unstable to the elliptic instability owing to the elliptic deformation of their core. These elliptic instability modes persist for weakly stratified flow: $F_h \,{>}\, 10$, where $F_h$ is the horizontal Froude number ($F_h\,{=}\,\Gamma_b/\pi a_b^2 N$ where $\Gamma_b$ is the circulation of the vortices, $a_b$ their core radius and $N$ the Brunt–Väisälä frequency). For strong stratification ($F_h \,{<}\, 2.85$), a new zigzag instability is found that bends each vortex symmetrically with almost no internal deformation of the basic vortices. This instability may modify the vortex merging since at every half-wavelength along the vertical, the vortices are alternatively brought closer, accelerating the merging, and moved apart, delaying the merging. The most unstable vertical wavelength $\lambda_m$ of this new instability is shown to be proportional to $F_h b_b$, where $b_b$ is the distance between the vortices, implying that $\lambda_m$ decreases with increasing stratification. The maximum growth rate, however, is independent of the stratification and proportional to the strain $S\,{=}\,\Gamma_b/2 \pi b_b^2 $. These scaling laws and the bending motion induced by this instability are similar to those of the zigzag instability of a counter-rotating vortex pair in a stratified fluid.