Temporal Growth Analysis for Prandtl Slope Flows: Modal or Non-modal?

In 1942, Ludwig Prandtl introduced a simple 1D model to study stably stratified flows over sloped terrain such as nocturnal mountain or valley winds. It assumes a constant background stratification as well as uniform surface cooling on an infinite slope and has served as a canonical problem to enhance our understanding of stably stratified flows over non-flat surfaces. Prior investigations into the stability of Prandtl slope flows have indicated that the results from linear modal analysis were able to accurately capture the growth dynamics of small initial perturbations to the 1D laminar Prandtl slope profiles. This suggests that, in contrast to shear-driven flows, the linearized Navier-Stokes operator derived from Prandtl's base flow profile exhibits a large degree of normality, thus inhibiting strong transient growth rates that can dominate the modal growth. We present results from non-modal analysis over a large range of slope angles for both anabatic and katabatic slope flows to gather quantitative evidence for this assertion, which will help explain which dynamics are most significant during the transition to instability and turbulence in such flows. In a broader context, our results also help understand whether, despite its simplicity, linear modal analysis could be an adequate tool to comprehend the instability dynamics of Prandtl slope flows and related flow problems under the right circumstances.