The effects of Prandtl number on the nonlinear dynamics of Kelvin-Helmholtz instability in two dimensions

It is known that the pitchfork bifurcation of Kelvin-Helmholtz instability occurring at minimum gradient Richardson number $Ri_m \simeq 1/4$ in viscous stratified shear flows can be subcritical or supercritical depending on the value of the Prandtl number, $Pr$. Here we study stratified shear flow restricted to two dimensions at finite Reynolds number, continuously forced to have a constant background density gradient and a hyperbolic tangent shear profile, corresponding to the `Drazin model' base flow. Bifurcation diagrams are produced for fluids with $Pr=0.7$ (typical for air), 3 and $7$ (typical for water). For $Pr=3$ and $7$, steady billow-like solutions are found to exist for strongly stable stratification of $Ri_m$ up to $1/2$ and beyond. Interestingly, these solutions are not a direct product of a Kelvin-Helmholtz instability having too short a wavelength but can give rise to Kelvin-Helmholtz states of twice the wavelength through subharmonic bifurcations. These short-wavelength states can be tracked down to at least $Pr \approx 2.3$ and act as instigators of complex dynamics even in strongly stratified flows when the flow is unforced.