Two pathways to diapycnal mixing in strongly stratified flows with no initial vertical shear

Abstract

While vertically-sheared stratified flows have been studied extensively, their horizontally-sheared counterparts have received considerably less attention. Yet, horizontal shear instabilities remain active even when the mean Richardson number is large or even formally infinite, and can drive turbulence in strongly stratified (low Froude number) flows at sufficiently high Reynolds number. In this work, we combine linear theory with direct numerical simulations to investigate two pathways to turbulence in low Froude / high Reynolds number horizontally-sheared flow with no initial vertical shear. In the first pathway, vertical shear emerges directly from vertically-modulated eigenmodes of the primary horizontal shear instability, and becomes unstable to secondary small-scale Kelvin-Helmholtz (KH) instabilities on the buoyancy scale at sufficiently large buoyancy Reynolds number $Re_b$. In the second pathway, a vertically-invariant eigenmode of the primary horizontal shear instability initially dominates, causing the background flow to evolve nonlinearly into a long-lived time-dependent two-dimensional (columnar) vortical flow. The vortices are subsequently unstable to secondary three-dimensional hyperbolic instabilities from which vertical shear emerges, which is finally unstable to tertiary small-scale KH instabilities on the buoyancy scale at sufficiently large $Re_b$. This shows that the emergence of vertical shear driving small-scale KH instabilities is an inevitable by-product of horizontal shear instabilities in strongly stratified flows at sufficiently large $Re_b$. However, we also find that the two pathways excite different ranges of vertical scales, which results in different peak mixing efficiencies.