Onset of secondary instabilities on the zigzag instability in stratified fluids

Recently, Deloncle, Billant & Chomaz (J. Fluid Mech., vol. 599, 2008, p. 229) and Waite & Smolarkiewicz (J. Fluid Mech., vol. 606, 2008, p. 239) have performed numerical simulations of the nonlinear evolution of the zigzag instability of a pair of counter-rotating vertical vortices in a stratified fluid. Both studies report the development of a small-scale secondary instability when the vortices are strongly bent if the Reynolds number Re is sufficiently high. However, the two papers are at variance about the nature of this secondary instability: it is a shear instability according to Deloncle et al. (J. Fluid Mech., vol. 599, 2008, p. 229) and a gravitational instability according to Waite & Smolarkiewicz (J. Fluid Mech., vol. 606, 2008, p. 239). They also profoundly disagree about the condition for the onset of the secondary instability: Re F-h(2) > O(1) according to the former or Re F-h > 80 according to the latter, where F-h is the horizontal Froude number. In order to understand the origin of these discrepancies, we have carried out direct numerical simulations of the zigzag instability of a Lamb-Chaplygin vortex pair for a wide range of Reynolds and Froude numbers. The threshold for the onset of a secondary instability is found to be Re F-h(2) similar or equal to 4 for Re greater than or similar to 3000 and Re F-h = 80 for Re less than or similar to 1000 in agreement with both previous studies. We show that the scaling analysis of Deloncle et al. (J. Fluid Mech., vol. 599, 2008, p. 229) can be refined to obtain a universal threshold: (Re - Re-0)F-h(2) similar or equal to 4, with Re-0 similar or equal to 400, which works for all Re. Two different regimes for the secondary instabilities are observed: when (Re - Re-0)F-h(2) similar or equal to 4, only the shear instability develops while when (Re - Re-0)F-h(2) >> 4, both shear and gravitational instabilities appear almost simultaneously in distinct regions of the vortices. However, the shear instability seems to play a dominant role in the breakdown into small scales in the range of parameters investigated.