THREE-DIMENSIONAL SLOPE FLOWS IN A STRATIFIED FLUID

我们考虑一个平面倾斜的边界，浸入密度/热分层的流体中。如果斜坡相对于相同水平面上的分层环境流体是冷的，那么就会出现密度更大的斜坡附近流体的下坡“倒转”流动。这种下坡流通常在夜间斜坡地形冷却的环境应用中发现。这项工作探索了这些流动的特征，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的建模，计算，流体动力稳定性理论和渐近方法的结合。从现有的一维变流普朗特模型出发，分析了该普朗特基流的线性稳定性。发现并分析了两种不同类型的扰动的参数范围，即沿坡排列的涡（旋涡模式）和沿坡传播的波（波模式）。为了与倾斜流的主要地球物理应用保持一致，我们将重点放在浅斜坡角（<5度）上。我们表明，漩涡模式的不稳定性通常在浅坡角处占主导地位，因此具有特殊的实际意义，但会导致完全三维的流动，这与以前的简单建模有很大的不同。我们通过探索(a)普朗特基流的弱非线性稳定性和(b)计算的完全非线性涡旋状态来考虑通过这种不稳定性可能产生的非线性（稳态）涡旋状态。最后，我们开始通过在旋转参考系中分析系统来解释地球的旋转效应，从而包括科里奥利力。对于较慢的下坡流动，这些旋转效应变得更加重要。

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.