A Lagrangian view of mixing in stratified shear flows

We consider numerically the effect of turbulent mixing in stably stratified parallel shear flow where the initial velocity and density have a hyperbolic tangent profile in the vertical coordinate with the same inflection point. The flow can be susceptible to two types of instabilities: Kelvin-Helmholtz instability (KHI) and Holmboe wave instability (HWI). These instabilities lead to two distinct types of mixing: mixing by ‘overturning’ and mixing by ‘scouring’. We examine mixing from a Lagrangian perspective using direct numerical simulations (DNS). Lagrangian particles are tracked in the simulations, and the fluid buoyancy sampled along particle paths provides a Lagrangian measure of mixing. The particles exhibit aggregation in buoyancy space when there is sustained overturning motion within the interface. The root mean square buoyancy for a set of particles that start with the same buoyancy is larger for HWI than KHI for the same bulk Richardson number. Finally, 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 end on the opposite side of the mid-plane is compared for KHI and HWI for several values of the bulk Richardson number. Surprisingly, for HWI with a large bulk Richardson number, more than half of the particles that start near the mid-plane end on the opposite side of the mid-plane.