Nonlinear evolution of the zigzag instability in stratified fluids: a shortcut on the route to dissipation

Abstract

We present high-resolution direct numerical simulations of the nonlinear evolution of a pair of counter-rotating vertical vortices in a stratified fluid for various high Reynolds numbers Re and low Froude numbers Fh. The vortices are bent by the zigzag instability producing high vertical shear. There is no nonlinear saturation so that the exponential growth is stopped only when the viscous dissipation by vertical shear is of the same order as the horizontal transport, i.e. when $Z^h_{\hbox{\it\scriptsize max}}$/Re=O(1) where $Z^h_{\hbox{\it\scriptsize max}}$ is the maximum horizontal enstrophy non-dimensionalized by the vortex turnover frequency. The zigzag instability therefore directly transfers the energy from large scales to the small dissipative vertical scales. However, for high Reynolds number, the vertical shear created by the zigzag instability is so intense that the minimum local Richardson number Ri decreases below a threshold of around 1/4 and small-scale Kelvin–Helmholtz instabilities develop. We show that this can only occur when $ReF_h^2$ is above a threshold estimated as 340. Movies are available with the online version of the paper.