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

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 = Γ_h/πa_b² N where Γ_h 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 λ_m of this new instability is shown to be proportional to F_hb_b, where b_b is the distance between the vortices, implying that λ_m decreases with increasing stratification. The maximum growth rate, however, is independent of the stratification and proportional to the strain S = Γ_b/2πb_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.