Theoretical analysis of the zigzag instability of a vertical columnar vortex pair in a strongly stratified fluid

A general theoretical account is proposed for the zigzag instability of a vertical columnar vortex pair recently discovered in a strongly stratified experiment. The linear inviscid stability of the Lamb-Chaplygin vortex pair is analysed by a multiple-scale expansion analysis for small horizontal Froude number (F-h = U/LhN, where U is the magnitude of the horizontal velocity, L-h the horizontal lengthscale and N the Brunt-Vaisala frequency) and small vertical Froude number (F-v = U/LvN, where L-v is the vertical lengthscale) using the scaling of the equations of motion introduced by Riley, Metcalfe & Weissman (1981). In the limit F-v = 0, these equations reduce to two-dimensional Euler equations for the horizontal velocity with undetermined vertical dependence. Thus, at leading order, neutral modes of the how are associated, among others, to translational and rotational invariances in each horizontal plane. To each broken invariance is related a phase variable that may vary freely along the vertical. Conservation of mass and potential vorticity impose at higher order the evolution equations governing the phase variables that we derive for F-h much less than 1 and F-v much less than 1 in the spirit of phase dynamics techniques established for periodic patterns. In agreement with the experimental observations, this asymptotic analysis shows the existence of an instability consisting of a vertically modulated rotation and a translation of the columnar vortex pair perpendicular to the travelling direction. The dispersion relation as well as the spatial eigenmode of the zigzag instability are determined. The analysis predicts that the most amplified vertical wavelength should scale as U/N and the maximum growth rate as U/L-h. Our main finding is thus that the typical thickness of the ensuing layers will be such that F-v = 0(1) and not F-v much less than 1 as assumed by Riley et al. (1981) and Lilly (1983). This implies that such strongly stratified flows are not described by two-dimensional horizontal equations. These results may help to understand the layering commonly observed in stratified turbulence and the fundamental differences with strictly two-dimensional turbulence.