We investigate the stability of katabatic slope flows over an infinitely wide and uniformly cooled planar surface subject to an additional forcing due to a uniform downslope wind field aloft. We adopt an extension of Prandtl's original model for slope flows (Lykosov & Gutman 1972) to derive the base flow, which constitutes an interesting basic state in stability analysis because it cannot be reduced to a single universal form independent of external parameters. We apply a linear modal analysis to this basic state to demonstrate that for a fixed Prandtl number and slope angle, two independent dimensionless parameters are sufficient to describe the flow stability. One of these parameters is the stratification perturbation number that we have introduced in Xiao & Senocak (2019). The second parameter, which we will henceforth designate the wind forcing number, is hitherto uncharted and can be interpreted as the ratio of the kinetic energy of the ambient wind aloft to the damping due to viscosity and stabilizing effect of the background stratification. For a fixed Prandtl number, stationary transverse and travelling longitudinal modes of instabilities can emerge, depending on the value of the slope angle and the aforementioned dimensionless numbers. The influence of ambient wind forcing on the base flow's stability is complicated as the ambient wind can be both stabilizing as well as destabilizing for a certain range of the parameters. Our results constitute a strong counter-evidence against the current practice of relying solely on the gradient Richardson number to describe the dynamic stability of stratified atmospheric slope flows.

中文翻译：

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随环境风力变化的 Prandtl 斜坡流的线性稳定性

我们研究了在无限宽且均匀冷却的平面上的下降坡流的稳定性，该表面受到由于高空均匀下坡风场而产生的额外强迫。我们采用 Prandtl 的斜坡流原始模型 (Lykosov & Gutman 1972) 的扩展来推导基流，它构成稳定性分析中一个有趣的基本状态，因为它不能被简化为独立于外部参数的单一通用形式。我们将线性模态分析应用于这种基本状态，以证明对于固定的普朗特数和倾斜角，两个独立的无量纲参数足以描述流动稳定性。这些参数之一是我们在 Xiao & Senocak (2019) 中引入的分层扰动数。第二个参数，我们将在此之后指定风力数，它迄今尚未被绘制出来，可以解释为高空环境风的动能与由于背景分层的粘性和稳定效应引起的阻尼之比。对于固定的 Prandtl 数，可能会出现平稳的横向和纵向不稳定模式，这取决于倾斜角的值和上述无量纲数。环境风强迫对基流稳定性的影响是复杂的，因为环境风对于一定范围的参数既可以是稳定的，也可以是不稳定的。我们的结果有力地反驳了目前仅依靠梯度理查森数来描述分层大气坡流的动态稳定性的做法。