THREE-DIMENSIONAL SLOPE FLOWS IN A STRATIFIED FLUID

我们考虑一个平面倾斜边界，浸没在密度/热分层的流体中。如果斜坡相对于同一水平线上的分层环境流体是冷的，那么可以预料到更稠密的斜坡附近流体的下坡式‘katabatic’流动。这种下坡流通常出现在坡地夜间降温的环境应用中。这项工作探索了这种流动的特征，当它们是三维的，与一系列现有的和更简单的模型形成对比。我们的方法是理想化模型、计算、流体动力稳定性理论和渐近方法的结合。从现有的一维Prandtl流动模型出发，分析了这种Prandtl基流的线性稳定性。对于两种不同类型的扰动，分别对应于沿斜坡排列的涡旋(涡模)和向下传播的波(波模)，找到并分析了区分稳定区和不稳定区的参数范围。为了与绝热气流的主要地球物理应用保持一致，我们将重点放在浅坡角(&lt；5度)上。我们发现，涡模不稳定性通常在浅坡角处占主导地位，因此具有特殊的实用价值，但会导致完全三维的流动，这与以前的简单模拟有很大不同。通过研究(A)Prandtl基流的弱非线性稳定性和(B)完全非线性涡旋态的计算，我们考虑了由这种不稳定性可能产生的非线性(定常)涡旋态。最后，我们通过分析旋转坐标系中的系统，从而包括科里奥利力，开始考虑地球的自转效应。这些旋转效应对于较慢的下坡流来说变得更加重要。

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.