  Highly stratified shear layers are rendered unstable even at high
stratifications by Holmboe instabilities when the density stratification is
concentrated in a small region of the shear layer. These instabilities may
cause mixing in highly stratified environments. However these instabilities
occur in limited bands in the parameter space. We perform Generalized Stability
analysis of the two dimensional perturbation dynamics of an inviscid Boussinesq
stratified shear layer and show that Holmboe instabilities at high Richardson
numbers can be excited by their adjoints at amplitudes that are orders of
magnitude larger than by introducing initially the unstable mode itself. We
also determine the optimal growth that is obtained for parameters for which
there is no instability. We find that there is potential for large transient
growth regardless of whether the background flow is exponentially stable or not
and that the characteristic structure of the Holmboe instability asymptotically
emerges as a persistent quasi-mode for parameter values for which the flow is
stable.
