A Lagrangian view of mixing in stratified shear flows

Stratified shear flows are commonplace in the ocean and the atmosphere. Understanding the mechanisms by which such flows become turbulent and lead to irreversible mixing due to the ultimate break down of different types of primary instabilities is vital in understanding diapycnal fluxes of heat and other important scalars such as salt and carbon. We consider numerically the Lagrangian view of turbulent mixing in stably stratified parallel shear flow where both the initial velocity field and initial density departure from the base hydrostatic state have a hyperbolic tangent profile in the vertical coordinate with the same point of inflection. By varying the ratio of velocity interface thickness and density interface thickness, these initial conditions permit two types of instabilities: Kelvin-Helmholtz instability (KHI) and Holmboe wave instability (HWI). These instabilities lead to two distinct types of mixing; overturning motions within the density interface, and scouring by turbulence on the edges of the density interface. Here, we examine mixing from a Lagrangian perspective using direct numerical simulations (DNS) for initial conditions that are unstable to KHI and HWI. Lagrangian particles are tracked in the simulations, and the fluid buoyancy sampled along particle paths provides a Lagrangian measure of mixing. The timing of mixing events experienced by particles inside and outside the interface is different in simulations exhibiting KHI and HWI. The particles exhibit aggregation in buoyancy space when there is sustained overturning motion within the interface. The root mean square (RMS) buoyancy for a set of particles that start with the same buoyancy is larger for HWI than KHI for the same bulk Richardson number, implying heterogeneous mixing along particle paths for HWI. 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. We explain this result in terms of localisation of mixing.