THREE-DIMENSIONAL SLOPE FLOWS IN A STRATIFIED FLUID

We consider a planar sloping boundary, immersed in a fluid that is density/thermally stratified. If the slope is cold relative to the stratified ambient fluid at the same horizontal level, then a down-slope 'katabatic' flow of denser slope-adjacent fluid is to be expected. Such downslope flows are commonly found in environmental applications with night-time cooling of sloping terrains. This work explores the characteristics of such flows when they are three-dimensional, in contrast to a range of existing and more simplistic models. Our approach is a combination of idealised modelling, computation, hydrodynamic stability theory and asymptotic methods.

Starting from the existing one-dimensional Prandtl model for katabatic flows, the linear stability of this Prandtl baseflow is analysed. The parameter ranges that delineate regions of stability from regions of instability are found and analysed for two distinct classes of disturbance, corresponding to slope-aligned vortices (vortex modes) and down-slope propagating waves (wave modes). To remain consistent with the dominant geophysical applications of katabatic flows, we focus on shallow slope angles (<5 degrees). We show that the vortex mode instabilities typically dominate at shallow slope angles and are therefore of particular practical interest, but lead to fully three-dimensional flows that depart significantly from previous simple modelling. We consider nonlinear (steady) vortex states that can arise via this instability by exploring (a) the weakly nonlinear stability of the Prandtl baseflow and (b) the fully nonlinear vortex states computationally.

Finally, we begin to account for the rotational effects of the Earth by analysing the system in a rotating frame of reference, thereby including Coriolis forcing. These rotational effects become more important for slower down-slope flows.