Instability and breakdown of a vertical vortex pair in a strongly stratified fluid

The dynamics of a counter-rotating pair of columnar vortices aligned parallel to a stable density gradient are investigated. By means of large-eddy and direct numerical simulations, we extend the linear analyses and laboratory experiments of Billant and Chomaz (thoroughly documented in three articles in J. Fluid Mech. in 2000) to the fully nonlinear, large-Reynolds-number regime. A range of stratifications and vertical length scales is considered, with Fr_h < 0.2 and 0.1 < Fr_z < 10. Here Fr_h = U/(NR) and Fr_z = Uk_z/N are the horizontal and vertical Froude numbers, U and R are the horizontal velocity and length scales of the vortices, N is the Brunt-Väisälä frequency, and k_z is the vertical wavenumber of a small initial perturbation. At early times with Fr_z < 1, linear predictions for the zigzag instability are reproduced. Short-wavelength perturbations with Fr_z > 1 are found to be unstable as well, with growth rates only slightly less than those of the zigzag instability but with very different structure. At later times, the large-Reynolds-number evolution diverges profoundly from the laboratory experiments as the instabilities transition to turbulence. For the zigzag instability, this transition occurs when density perturbations generated by the vortex bending become gravitationally unstable. The resulting turbulence rapidly destroys the vortex pair. We derive the criterion η/R = 0.2/Fr_z for the onset of gravitational instability, where η is the maximum horizontal displacement of the bent vortices; our simulations agree for the fastest growing wavelengths 0.3 < Fr_z < 0.8. Short perturbations with Fr_z > 1 saturate at low amplitude, preserving the columnar structure of the vortices well after the generation of turbulence. Viscosity is shown to suppress the transition to turbulence, yielding laminar dynamics and, under certain conditions, pancake vortices like those observed in the laboratory.