A Lagrangian view of mixing in stratified shear flows

We consider numerically a Lagrangian view of turbulent mixing in two-layer stably stratified parallel shear flow. By varying the ratio of shear layer depth to density interface thickness, these flows are prone to either a primary Kelvin-Helmholtz instability (KHI) or to a primary Holmboe wave instability (HWI). These instabilities are conventionally thought to mix qualitatively differently; by vortical `overturning' of the density interface induced by KHI, or by turbulent `scouring' on the edges of the density interface induced by HWI. By tracking Lagrangian particles in direct numerical simulations, so that the fluid buoyancy sampled along particle paths provides a particular Lagrangian measure of mixing, we investigate the validity of this overturning/scouring classification. The timing of mixing events experienced by particles inside and outside the interface is qualitatively different in simulations exhibiting KHI and HWI. The root mean square (RMS) buoyancy for particles that start with the same buoyancy is actually larger for HWI-associated flows than for KHI-associated flows for the same bulk Richardson number Ri_b, implying heterogeneous mixing along particle paths for HWI. The number of particles starting close to the mid-plane of the interface which experience a change in sign in the local fluid buoyancy (and hence end up on the opposite side of the mid-plane after mixing) is compared for KHI and HWI in flows with various Ri_b. Perhaps surprisingly, for HWI with a large Ri_b, more than half of the particles that start near the mid-plane end up on the opposite side of the mid-plane.