Weakly nonlinear Holmboe waves

Holmboe waves are long-lived traveling waves commonly found in environmental stratified shear flows. Here we study their finite-amplitude properties in the nonlinear but nonturbulent regime, with a weakly nonlinear temporal stability analysis. Using a versatile amplitude expansion method, we analyze supercritical bifurcation diagrams both in Reynolds number and Richardson number, transient phase portraits, and the vertical structures of all components modes up to third order. We believe these results provide a basis for a future fully nonlinear analysis of the Holmboe dynamical system.