  We present a numerical investigation of three-dimensional, short-wavelength
linear instabilities in Kelvin-Helmholtz (KH) vortices in homogeneous and
stratified environments. The base flow, generated using two-dimensional
numerical simulations, is characterized by the Reynolds number and the
Richardson number defined based on the initial one-dimensional velocity and
buoyancy profiles. The local stability equations are then solved on closed
streamlines in the vortical base flow, which is assumed quasi-steady. For the
unstratified case, the elliptic instability at the vortex core dominates at
early times, before being taken over by the hyperbolic instability at the
vortex edge. For the stratified case, the early time instabilities comprise a
dominant elliptic instability at the core and a hyperbolic instability strongly
influenced by stratification at the vortex edge. At intermediate times, the
local approach shows a new branch of instability (convective branch) that
emerges at the vortex core and subsequently moves towards the vortex edge. A
few more convective instability branches appear at the vortex core and move
away, before coalescing to form the most unstable region inside the vortex
periphery at large times. The dominant instability characteristics from the
local approach are shown to be in good qualitative agreement with results from
global instability studies for both homogeneous and stratified cases.
Compartmentalized analyses are then used to elucidate the role of shear and
stratification on the identified instabilities. The role of buoyancy is shown
to be critical after the primary KH instability saturates, with the dominant
convective instability shown to occur in regions with the strongest statically
unstable layering. We conclude by highlighting the potentially insightful role
that the local approach may offer in understanding the secondary instabilities
in other flows.
