Shallow katabatic flow on a non-uniformly cooled slope

We examine katabatic flow driven by a non-uniformly cooled slope surface but unaffected by Coriolis acceleration. A general formulation is given, valid for non-uniform surface buoyancy distributions over a down-slope length scale L≫δ0, where δ0=ν/(Nsinα)1/2 is the slope-normal Prandtl depth, for a kinematic viscosity ν, buoyancy frequency N and slope angle α. We demonstrate that the similarity solution of Shapiro and Fedorovich (J Fluid Mech 571:149–175, 2007) can remain quantitatively relevant local to the end of a non-uniformly cooled region. The usefulness of the steady similarity solution is determined by a spatial eigenvalue problem on the L length scale. Broadly speaking, there are also two modes of temporal instability; stationary down-slope aligned vortices and down-slope propagating waves. By considering the limiting inviscid stability problem, we show that the origin of the vortex mode is spatial oscillation of the buoyancy profile normal to the slope. This leads to vortex growth in a region displaced from the slope surface, at a point of buoyancy inflection, just as the propagating modes owe their existence to an inflectional velocity. Non-uniform katabatic flows that detrain fluid to the ambient are shown to further destabilise the vortex mode whereas entraining flows lead to weaker vortex growth rates. Rayleigh waves dominate in general, but the vortex modes become more significant at small slope angles and we quantify their relative growth rates.