The dynamics of stacked density-stratified shear layers

Geophysical flows sometimes contain multiple vertically stacked stratified shear layers at close proximity to each other. Interactions between neighbouring layers can excite new instabilities and have non-trivial effects on the ensuing turbulence. We therefore consider the stability of a 2D stably stratified three-layer fluid. The fluid is sheared symmetrically over a distance 2h across each density interface and the shear layers are separated by a quiescent region of depth 2H. In the limits of zero and infinite shear layer separation, we recover two classic configurations. When H/h = 0, our basic state reduces to a single shear layer encompassing both density interfaces (in which both Holmboe wave instabilities and the Taylor-Caulfield instability are predicted to arise for sufficiently strong stratification). On the other hand, layer interaction effects weaken as H/h → ∞, implying only independent Holmboe wave instabilities localised in each layer. At finite, but small, H/h, inviscid linear stability analysis reveals the existence of new instabilities and regime transitions relative to the zero-separation case. We explore the effects of diffusion on these new features and study their nonlinear evolution using direct numerical simulations.