THREE-DIMENSIONAL SLOPE FLOWS IN A STRATIFIED FLUID

我们考虑一个平面倾斜的边界，沉浸在流体的密度/热分层。如果斜坡相对于同一水平高度处的分层环境流体是冷的，则预期斜坡附近的密度更大的流体的下坡“下降”流。这种下坡流在斜坡地形夜间冷却的环境应用中很常见。这项工作探讨了这种流动的特点时，他们是三维的，在现有的范围内，更简单的模型。我们的方法是理想化模型、计算、流体动力学稳定性理论和渐近方法的结合，从现有的下降流的一维Prandtl模型出发，分析了这个Prandtl基流的线性稳定性。的参数范围，划定区域的稳定性不稳定的区域被发现和分析两个不同的类的干扰，对应于斜坡对齐的涡（涡模式）和下坡传播波（波模式）。为了保持与下降流的主要地球物理应用的一致性，我们专注于浅坡角（<5度）。我们表明，涡模式的不稳定性通常占主导地位，在浅的倾斜角，因此特别实际的利益，但导致完全三维的流动，从以前的简单建模显着离开。我们考虑非线性（稳定）涡状态，可以通过这种不稳定性，探索（a）弱非线性稳定性的普朗特基流和（B）完全非线性涡状态计算。最后，我们开始考虑地球的旋转效应，通过分析旋转参考系中的系统，从而包括科里奥利强迫。这些旋转效应对于较慢的下坡流变得更加重要。

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.