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
(Fh = U/LhN,
where U is the magnitude of the horizontal velocity, Lh the horizontal lengthscale
and N the Brunt–Väisälä frequency) and small vertical Froude number
(Fv = U/LvN,
where Lv is the vertical lengthscale) using the scaling of the equations of motion
introduced by Riley, Metcalfe & Weissman (1981). In the limit Fv = 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 flow
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
Fh [Lt ] 1 and Fv [Lt ] 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/Lh.Our main finding is thus that the typical thickness of the ensuing layers will be
such that Fv = O(1) and not Fv [Lt ] 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.
SPATIOTEMPORAL PERIODIC PATTERNS OF A TWO-DIMENSIONAL SYMMETRICALLY COUPLED MAP LATTICES
