Secondary Instabilities of Longitudinal Rolls in Anabatic Prandtl Slope Flows

Stationary longitudinal rolls arise as nonlinear saturation of a linear instability in Prandtl's laminar anabatic flows at small slope angles when the ratio between imposed surface buoyancy gradient and ambient stratification, a dimensionless number designated as the stratification perturbation number, becomes sufficiently large. In the present work, the instability modes of these slope-aligned vortices are investigated with the help of bi-global linear stability analysis. We identify multiple possible dynamics depending on the separation of the original neighboring vortex pairs as well as the stratification perturbation number. It turns out that for almost all conditions, the most dominant modes are subharmonic, which are responsible for the anti-parallel distortion and merging of neighboring vortex pairs. For very specific initial vortex pair separations, the two-dimensional fundamental instability that is characteristic for parallel transverse vortex translation becomes most dominant. The dependence of  secondary instability growth rates and oscillations on the longitudinal wave number and the stratification perturbation number as well as the initial vortex pair separation distance will be presented. A remarkable feature of all the configurations studied herein is the absence of any dominant short-wave three-dimensional modes, which implies that the original vortices maintain their longitudinal structure even after the onset of secondary instability dynamics.