Pairing of two vertical columnar vortices in a stratified fluid
Résumé
We present three-dimensional (3D) numerical simulations of the pairing of two vertical columnar vortices in a stably stratified fluid. Whereas in two dimensions, merging of two isolated vortices occurs on a diffusion time scale, in the three-dimensional stratified case we show that merging is a much faster process that occurs over an inertial time scale. The sequence of dynamical processes that leads to this accelerated pairing involves first a linear stage where the zigzag instability develops displacing vortices alternately closer and farther with a vertical periodicity scaling on the buoyancy length scale LB=Fhb, where Fh is the horizontal Froude number (Fh=Γ/πa2N with a the core size of the vortices, Γ their circulation and N the Brunt–Väisälä frequency) and b is the separation distance between the vortices. In layers where the vortices have started to move closer, their distance decreases exponentially with the growth rate of the zigzag instability. Non-linearities do not seem to affect this process and the decrease only stops when the pairing is completed in that layer. At the same time, enstrophy that has also grown exponentially reaches a magnitude of the order of the Reynolds number Re=Γ/(πν) (where ν is the kinematic viscosity of the fluid) if the Reynolds number is not too large, meaning that energy is then dissipated on the inertial time scale. This dissipation occurs in thin layers and the vortices that were originally moving away in the intermediate layer start slowing down and rapidly merge.