The three-dimensional stability of vertical vortex pairs in stratified and rotating fluids is investigated using the analytical approach established in Part 1 and the predictions are compared to the results of previous direct numerical stability analyses for pairs of co-rotating equal-strength Lamb–Oseen vortices and to new numerical analyses for equal-strength counter-rotating vortex pairs. A very good agreement between theoretical and numerical results is generally found, thereby providing a comprehensive description of the zigzag instability. Co-rotating and counter-rotating vortex pairs are most unstable to the zigzag instability when the Froude number Fh = Γ/(2πR2N) (where Γ is the vortex circulation, R the vortex radius and N the Brunt–Väisälä frequency) is lower than unity independently of the Rossby number Ro = Γ/(4πR2Ωb) (Ωb is the planetary rotation rate). In this range, the maximum growth rate is proportional to the strain Γ/(2πb2) (b is the separation distance between the vortices) and is almost independent of Fh and Ro. The most amplified wavelength scales like Fhb when the Rossby number is large and like Fhb/|Ro| when |Ro| ≪ 1, in agreement with previous results. While the zigzag instability always bends equal-strength co-rotating vortex pairs in a symmetric way, the instability is only quasi-antisymmetric for finite Ro for equal-strength counter-rotating vortex pairs because the cyclonic vortex is less bent than the anticyclonic vortex. The theory is less accurate for co-rotating vortex pairs around Ro ≈ −2 because the bending waves rotate very slowly for long wavelength. The discrepancy can be fully resolved by taking into account higher-order three-dimensional effects.When Fh is increased above unity, the growth rate of the zigzag instability is strongly reduced because the bending waves of each vortex are damped by a critical layer at the radius where the angular velocity of the vortex is equal to the Brunt–Väisälä frequency. The zigzag instability, however, continues to exist and is dominant up to a critical Froude number, which mostly depends on the Rossby number. Above this threshold, equal-strength co-rotating vortex pairs are stable with respect to long-wavelength bending disturbances whereas equal-strength counter-rotating vortex pairs become unstable to a quasi-symmetric instability resembling the Crow instability in homogeneous fluids. However, its growth rate is lower than in homogeneous fluids because of the damping by the critical layer. The structure of the critical layer obtained in the computations is in excellent agreement with the theoretical solution.Physically, the different stability properties of vortex pairs in stratified and rotating fluids compared to homogeneous fluids are shown to come from the reversal of the direction of the self-induced motion of bent vortices.
Long-wavelength asymptotics of unstable crossflow modes, including the effect of surface curvature