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

Abstract

International audienceWe 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: Fh > 10, where Fh is the horizontal Froude number (Fh = Gh/pab2 N where Gh is the circulation of the vortices, ab their core radius and N the Brunt-Väisälä frequency). For strong stratification (Fh < 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 ?m of this new instability is shown to be proportional to Fhbb, where bb is the distance between the vortices, implying that ?m decreases with increasing stratification. The maximum growth rate, however, is independent of the stratification and proportional to the strain S = Gb/2pbb2. 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. © 2006 Cambridge University Press

The effect of planetary rotation on the zigzag instability of co-rotating vortices in a stratified fluid

“…The same dipolar shape in each vortex core with the same orientation (angle between the dipole axis and the line joining the vortices) is observed for all the Rossby numbers. As explained in Otheguy et al (2006) this shape corresponds to a bending of each vortex obliquely in opposite directions, with no deformation of their core. Together with the results found in Otheguy et al (2006) where the Rossby number was infinite and the Froude number was varied, from this we conclude that the self-similarity property of the zigzag instability on the single variable k z F h b/f (Ro) is therefore a fundamental property of the growth rate as well as the shape of the eigenmode of the instability.…”

“…The function f (Ro) is always real even for negative Ro. Alternatively, when the horizontal Froude number F h is varied, it has been shown in Otheguy et al (2006) that k zm ∝ 1/F h b when there is no rotation (Ro = ∞). Without further argument, these two scaling laws can be combined in the general form k zm = F (Ro,F h )/F h b where F is ap r i o r i a function of F h and Ro.…”

“…In a companion paper (Otheguy, Chomaz & Billant 2006), we have shown that such a basic flow in a strongly stratified fluid is affected by a zigzag instability which bends the two vortices symmetrically. In the non-rotating flow, the most unstable wavelength of this instability scales as the buoyancy length and its growth rate scales as the external strain that each vortex induces on the other one.…”

Short-wave vortex instability in stratified flow

“…The zigzag instability has a dominant vertical wavelength of around U/N , which is known as the buoyancy scale [18]. This instability has also been found in other flow configurations including co-rotating vortices [19] and vortex arrays [20]. The breakdown of this dipole into turbulence due to the growth and saturation of the zigzag instability has also been investigated [4,6,5].…”

Low Rossby limiting dynamics for stably stratified flow with finite Froude number

“…examine is Otheguy, Chomaz & Billant (2006). This work also uses a linear stability analysis but begins with an initial condition of co-rotating vortices.…”